ON THE ARGUMENT OF ABEL. William Rowan Hamilton

Transcription

1 ON THE ARGUMENT OF ABEL By Wllam Rowan Hamlton (Transactons of the Royal Irsh Academy, 8 (839), pp ) Edted by Davd R. Wlkns 2000

2 NOTE ON THE TEXT The text of ths edton s taken from the 8th volume of the Transactons of the Royal Irsh Academy. A small number of obvous typographcal errors have been corrected wthout comment n artcles, 3, 5, 6, 0 and 2. Davd R. Wlkns Dubln, February 2000

3 On the Argument of Abel, respectng the Impossblty of expressng a Root of any General Equaton above the Fourth Degree, by any fnte Combnaton of Radcals and Ratonal Functons. By Sr Wllam Rowan Hamlton. Read 22nd May, 837. [Transactons of the Royal Irsh Academy, vol. xv (839), pp ] [.] Let a, a 2,... a n be any n arbtrary quanttes, or ndependent varables, real or magnary, and let a, a 2,... a n be any n radcals, such that a α = f (a,... a n ),... a α n n = f n (a,... a n ); agan, let a,... a n be n new radcals, such that a α = f (a,... a n, a,... a n ), a α n n = f n (a,... a n, a,... a n ); and so on, tll we arrve at a system of equatons of the form a (m)α(m) = f (m ) (a (m ),... a (m ), a (m 2) n (m ),... a (m 2),... a n (m 2),... a n ), a (m)α(m) n (m) n (m) = f (m ) (a (m ) n (m),... a (m ), a (m 2) n (m ),... a (m 2),... a n (m 2),... a n ), the exponents α (k) beng all ntegral and prme numbers greater than unty, and the functons f (k ) beng ratonal, but all beng otherwse arbtrary. Then, f we represent by b (m) any ratonal functon f (m) of all the foregong quanttes a (k), b (m) = f (m) (a (m),... a (m) n (m), a (m ),... a (m ) n (m ),... a,... a n ), we may consder the quantty b (m) as beng also an rratonal functon of the n orgnal quanttes, a,... a n ; n whch latter vew t may be sad, accordng to a phraseology proposed by Abel, to be an rratonal functon of the m th order: and may be regarded as the general type of every concevable functon of any fnte number of ndependent varables, whch can be formed by any fnte number of addtons, subtractons, multplcatons, dvsons, elevatons

4 to powers, and extracton of roots of functons; snce t s obvous that any extracton of a radcal wth a composte exponent, such as α 2 α f, may be reduced to a system of successve extractons of radcals wth prme exponents, such as α f = f, α 2 f = f. Insomuch that the queston, Whether t be possble to express a root x of the general equaton of the n th degree, x n + a x n + + a n x + a n = 0, n terms of the coeffcents of that equaton, by any fnte combnaton of radcals and ratonal functons?, s, as Abel has remarked, equvalent to the queston, Whether t be possble to equate a root of the general equaton of any gven degree to an rratonal functon of the coeffcents of that equaton, whch functon shall be of any fnte order m? or to ths other queston: Is t possble to satsfy, by any functon of the form b (m), the equaton b (m)n + a b (m)n + + a n b (m) + a n = 0, n whch the exponent n s gven, but the coeffcents a, a 2,... a n are arbtrary? [2.] For the cases n = 2, n = 3, n = 4, ths queston has long snce been determned n the affrmatve, by the dscovery of the known solutons of the general quadratc, cubc, and bquadratc equatons. Thus, for n = 2, t has long been known that a root x of the general quadratc equaton, x 2 + a x + a 2 = 0, can be expressed as a fnte rratonal functon of the two arbtrary coeffcents a, a 2, namely, as the followng functon, whch s of the frst order: the radcal a beng such that x = b = f (a, a, a 2 ) = a 2 + a, a 2 = f (a, a 2 ) = a2 4 a 2; nsomuch that, wth ths form of the rratonal functon b, the equaton b 2 + a b + a 2 = 0 s satsfed, ndependently of the quanttes a and a 2, whch reman altogether arbtrary. Agan, t s well known that for n = 3, that s, n the case of the general cubc equaton x 3 + a x 2 + a 2 x + a 3 = 0, 2

5 a root x may be expressed as an rratonal functon of the three arbtrary coeffcents a, a 2, a 3, namely as the followng functon, whch s of the second order: x = b = f (a, a, a, a 2, a 3 ) = a 3 + a + c 2 a the radcal of hghest order, a, beng defned by the equaton a 3 = f (a, a, a 2, a 3 ) = c + a, and the subordnate radcal a beng defned by ths other equaton a 2 = f (a, a 2, a 3 ) = c 2 c 3 2, whle c and c 2 denote for abrdgment the followng two ratonal functons: c = 54 (2a3 9a a a 3 ), c 2 = 9 (a2 3a 2 ); so that, wth ths form of the rratonal functon b, the equaton b 3 + a b 2 + a 2 b + a 3 = 0 s satsfed, wthout any restrcton beng mposed on the three coeffcents a, a 2, a 3. For n = 4, that s, for the case of the general bquadratc equaton x 4 + a x 3 + a 2 x 2 + a 3 x + a 4 = 0, t s known n lke manner, that a root can be expressed as a fnte rratonal functon of the coeffcents, namely as the followng functon, whch s of the thrd order: x = b = f (a, a 2, a, a, a, a 2, a 3, a 4 ) = a 4 + a + a 2 + e 4 a ; a 2 ; wheren a 2 = f (a, a, a, a 2, a 3, a 4 ) = e 3 + a + e 2 a a 2 2 = f 2 (a, a, a, a 2, a 3, a 4 ) = e 3 + ρ 3 a + e 2 ρ 3 a a 3 = f (a, a, a 2, a 3, a 4 ) = e + a, a 2 = f (a, a 2, a 3, a 4 ) = e 2 e 3 2;,, 3

6 e 4, e 3, e 2, e denotng for abrdgment the followng ratonal functons: e 4 = 64 ( a3 + 4a a 2 8a 3 ), e 3 = 48 (3a2 8a 2 ), e 2 = 44 ( 3a a 3 + a a 4 ), e = 2 (3e 2e 3 e e 2 4) = 3456 (27a2 a 4 9a a 2 a 3 + 2a a 2 a a 2 3), and ρ 3 beng a root of the numercal equaton ρ ρ 3 + = 0. It s known also, that a root x of the same general bquadratc equaton may be expressed n another way, as an rratonal functon of the fourth order of the same arbtrary coeffcents a, a 2, a 3, a 4, namely the followng: x = b IV = f IV (a IV, a, a, a, a, a 2, a 3, a 4 ) = a 4 + a + a IV ; the radcal a IV beng defned by the equaton a IV2 = f (a, a, a, a, a 2, a 3, a 4 ) = a 2 + 3e 3 + 2e 4 whle a, a, a, and e 4, e 3, e 2, e, retan ther recent meanngs. Insomuch that ether the functon of thrd order b, or the functon of fourth order b IV, may be substtuted for x n the general bquadratc equaton; or, to express the same thng otherwse, the two equatons followng: b 4 + a b 3 + a 2 b 2 + a 3 b + a 4 = 0, and b IV4 + a b IV3 + a 2 b IV2 + a 3 b IV + a 4 = 0, are both dentcally true, n vrtue merely of the forms of the rratonal functons b and b IV, and ndependently of the values of the four arbtrary coeffcents a, a 2, a 3, a 4. But for hgher values of n the queston becomes more dffcult; and even for the case n = 5, that s, for the general equaton of the ffth degree, x 5 + a x 4 + a 2 x 3 + a 3 x 2 + a 4 x + a 5 = 0, the opnons of mathematcans appear to be not yet entrely agreed respectng the possblty or mpossblty of expressng a root as a functon of the coeffcents by any fnte combnaton of radcals and ratonal functons: or, n other words, respectng the possblty or mpossblty of satsfyng, by any rratonal functon b (m) of any fnte order, the equaton a b (m)5 + a b (m)4 + a 2 b (m)3 + a 3 b (m)2 + a 4 b (m) + a 5 = 0, 4,

7 the fve coeffcents a, a 2, a 3, a 4, a 5, remanng altogether arbtrary. To assst n decdng opnons upon ths mportant queston, by developng and llustratng (wth alteratons) the admrable argument of Abel aganst the possblty of any such expresson for a root of the general equaton of the ffth, or any hgher degree; and by applyng the prncples of the same argument, to show that no expresson of the same knd exsts for any root of any general but lower equaton, (quadratc, cubc, or bquadratc,) essentally dstnct from those whch have long been known; s the chef object of the present paper. [3.] In general, f we call an rratonal functon rreducble, when t s mpossble to express that functon, or any one of ts component radcals, by any smaller number of extractons of prme roots of varables, than the number whch the actual expresson of that functon or radcal nvolves; even by ntroducng roots of constant quanttes, or of numercal equatons, whch roots are n ths whole dscusson consdered as beng themselves constant quanttes, so that they nether nfluence the order of an rratonal functon, nor are ncluded among the radcals denoted by the symbols a, &c.; then t s not dffcult to prove that such rreducble rratonal functons possess several propertes n common, whch are adapted to assst n decdng the queston just now stated. In the frst place t may be observed, that, by an easy preparaton, the general rratonal functon b (m) of any order m may be put under the form b (m) = β (m) Σ <α (m) n whch the coeffcent b (m ) β (m),... β (m) n (m) exponent β (m) (m).(b (m ). a (m)β (m) β (m),... β (m)... a (m)β n (m) ), n (m) n (m) s a functon of the order m, or of a lower order; the s zero, or any postve nteger less than the prme number α (m), namely, as exponent nto the equaton of defnton of the radcal a (m) whch enters a (m)α(m) = f (m ) ; and the sgn of summaton extends to all the α (m).α (m) 2... α (m) β (m) terms whch have exponents n (m) subject to the condton just now mentoned. For, nasmuch as b (m) s, by supposton, a ratonal functon f (m) of all the radcals a (k) t s, wth respect to any radcal of hghest order, such as a (m), a functon of the form b (m) = n(a(m) ) m(a (m) ), m and n beng here used as sgns of some whole functons, or fnte ntegral polynomes. Now, f we denote by ρ α any root of the numercal equaton ρ (α ) α + ρ (α 2) α + ρ (α 3) α + + ρ 2 α + ρ α + = 0, 5,

8 so that ρ α s at the same tme a root of unty, because the last equaton gves ρ α α = ; and f we suppose the number α to be prme, so that ρ α, ρ 2 α, ρ 3 α,... ρ (α ) α are, n some arrangement or other, the α roots of the equaton above assgned: then, the product of all the α whole functons followng, m(ρ α a). m(ρ 2 αa)... m(ρ (α ) α a) = l(a), s not only tself a whole functon of a, but s one whch, when multpled by m(a), gves a product of the form l(a). m(a) = k(a α ), k beng here (as well as l) a sgn of some whole functon. If then we form the product m(ρ α (m) a (m) ). m(ρ 2 α (m) a (m) )... m(ρ α(m) α (m) a (m) ) = l(a (m) ), and multply, by t, both numerator and denomnator of the recently assgned expresson for b (m), we obtan ths new expresson for that general rratonal functon, b (m) = l(a(m) ). n(a (m) l(a (m) ) ). m(a (m) ) = l(a (m) ). n(a (m) k(a (m)α(m) ) ) = l(a(m) ). n(a (m) ) k(f (m ) ) = (a (m) ); the characterstc denotng here some functon, whch, relatvely to the radcal a (m), s whole, so that t may be thus developed, b (m) = (a (m) ) = 0 + a (m) + 2 a (m)2 + + r α (m)r, r beng a fnte postve nteger, and the coeffcents 0,,... r beng, n general, functons of the m th order, but not nvolvng the radcal a (m). And because the defnton of that radcal gves a (m)h = a (m)g. (f (m ) ) e, f h = g + eα (m), t s unnecessary to retan n evdence any of ts powers of whch the exponents are not less than α (m) ; we may therefore put the development of b (m) under the form b (m) = h 0 + h a (m) + + h α (m) 6 (a(m) ) α(m),

9 the coeffcents h 0, h,... beng stll, n general, functons of the m th order, not nvolvng the radcal a (m). It s clear that by a repetton of ths process of transformaton, the radcals a (m),... a (m) may all be removed from the denomnator of the ratonal functon f (m) ; and n (m) that ther exponents n the transformed numerator may all be depressed below the exponents whch defne those radcals: by whch means, the development above announced for the general rratonal functon b (m) may be obtaned; wheren the coeffcent b (m ) admts β (m),... β (m) n of beng analogously developed. (m) For example, the functon of the second order, b = a 3 + a + c 2 a whch was above assgned as an expresson for a root of the general cubc equaton, may be developed thus: n whch b = Σ.(b β <3 β b 0 = a 3, b =, b 2 = c 2 a 3. a β ) = b 0 + b a + b 2a 2 ;, = c 2 f = c 2 c + a. And ths last coeffcent b 2, whch s tself a functon of the frst order, may be developed thus: b 2 = c 2 c + a = b = Σ.(b β β. a β ) = b 0 + b a ; <2 n whch b 0 = c 2c c 2 = c 2c a 2 c 2 f Agan, the functon of the thrd order, = c 2c c 3 2 = c c 2 2 b = a 4 + a + a 2 + e 4 a, b = c 2. 2 a 2 whch expresses a root of the general bquadratc equaton, may be developed as follows:, n whch and b, = b = e 4 a 2. a 2 2 Σ.(b β <2 β 2 <2 β,β 2. a β. a β 2 2 ) = b 0,0 + b,0a + b 0,a 2 + b,a a 2 ; b 0,0 = a 4, b,0 =, b 0, =, = e 4 f. f 2 = ( = ( e 3 + ρ 2 e 3a + e ) 2 4 ρ 2. 3 a e 3 + a + e 2 a e 4 ) ( e 3 + ρ 3 a + e 2 ρ 3 a ) 7

10 And ths last coeffcent b,, whch s tself a functon of the second order, may be developed thus: b, = b = Σ.(b β <3 β. a β ) = b 0 + b a + b 2a 2 ; n whch b 0 = e 3, b = ρ2 3, b 2 = ρ 3e 2 ρ 3 e 2 e 4 e 4 e 4 a 3 = e 4 (e + a ) = ρ 3(e a ) e 4 e 2. 2 So that, upon the whole, these functons b and b, whch express, respectvely, roots of the general cubc and bquadratc equatons, may be put under the followng forms, whch nvolve no radcals n denomnators: and b = a ( ) a 3 + a + (c a 2 ), c 2 { b = a 4 + a + a 2 + ( ) } a e 3 + ρ 2 e 3a + ρ 3 (e a 2 ) a a 2 ; 4 e 2 and the functons f, f 2, whch enter nto the equatons of defnton of the radcals a, a 2, namely nto the equatons a 2 = f, a 2 2 = f 2, may n lke manner be expressed so as to nvolve no radcals n denomnators, namely thus: ( ) a a 2 = e 3 + a + (e a 2 ), ( ) a a 2 2 = e 3 + ρ 3 a + ρ 2 3(e a 2 ). e 2 It would be easy to gve other nstances of the same sort of transformaton, but t seems unnecessary to do so. [4.] It s mportant n the next place to observe, that any term of the foregong general development of the general rratonal functon b (m), may be solated from the rest, and expressed separately, as follows. Let b (m) denote a new rratonal functon, whch s γ (m),... γ (m) n (m) formed from b (m) by changng every radcal such as a (m) to a correspondng product such as ρ γ(m) α (m) a (m), n whch ρ (m) α b (m) γ (m),... γ (m) β (m) <α (m) n (m) = Σ e 2 s, as before, a root of unty; so that.(b (m ) β (m),... β (m) n (m). ρ β (m) γ (m) α (m)... ρ β(m) γ(m) n (m) n (m) α (m) n (m). a (m)β (m) and let any solated term of the correspondng development of b (m) or b (m) 0,... 0 the symbol t (m) β (m),... β (m) n (m) = b (m ). a (m)β (m) β (m),... β (m)... a (m)β n (m) 8 (m)... a (m)β n (m) ); n (m) (m) n (m) n (m) ; be denoted by

11 we shall then have, as the announced expresson for the solated term, the followng: t (m) β (m),... β (m) n (m) =. α (m)... α (m) Σ n (m) γ (m) <α (m).(b (m) γ (m),... γ (m) α (m). ρ β n (m) (m) γ (m)... ρ β(m) γ(m) n (m) n (m) ); α (m) n (m) the sgn of summaton here extendng to all those terms n whch every ndex such as γ (m) s equal to zero or to some postve nteger less than α (m). Thus, n the case of the functon of second order b, whch represents, as we have seen, a root of the general cubc equaton, f we wsh to obtan an solated expresson for any term of ts development already found, namely the development t β b = Σ.(b β <3 β we have only to ntroduce the functon b γ = Σ.(b β <3 β and to employ the formula t β In partcular, n whch = b β. a β = 3. Σ.(b γ <3 and n whch t s to be remembered that. a β ) = b 0 + b a + b 2a 2 = t 0 + t + t 2,. ρ β γ 3. a β ) = b 0 + b ρ γ γ 3 a + b 2ρ 2γ 3 b 3 a 2,. ρ β γ 3 ) = 3 (b 0 + ρ β + ρ 2β 2). t 0 = b 0 = 3 (b 0 + b + b 2), t = b a = 3 (b 0 + ρ 3 b + ρ 2 3 b 2), t 2 = b 2a 2 = 3 (b 0 + ρ 2 3 b + ρ 4 3 b 2), b 0 = b 0 + b a + b 2a 2 (= b ), b = b 0 + b ρ 3 a + b 2ρ 2 3a 2, b 2 = b 0 + b ρ 2 3a + b 2ρ 4 3a 2, ρ ρ 3 + = 0, and therefore ρ 3 3 =. 3 b Agan, f we wsh to solate any term t β of the development above assgned for the,β 2 functon of thrd order b, whch represents a root of the general bquadratc equaton, we may employ the formula t β,β 2 = b β,β 2. a β. a β 2 2 = 2.2. Σ γ <2 γ 2 <2.(b γ,γ 2. ρ β γ 2. ρ β = 4 {b 0,0 + ( ) β b,0 + ( ) β 2 b 0, + ( ) (β +β 2 ) b,}; 9 2 γ 2 2 )

12 n whch we have ntroduced the functon b γ,γ 2 = Σ.(b β <2 β 2 <2 β,β 2. ρ β γ 2. ρ β 2 γ 2 2. a β. a β 2 2 ) = b 0,0 + ( ) γ b,0a + ( ) γ 2 b 0,a 2 + ( ) γ +γ 2 b,a a 2 ; so that, n partcular, we have the four expressons n whch t 0,0 = b 0,0 = 4 (b 0,0 + b,0 + b 0, + b,), t,0 = b,0a = 4 (b 0,0 b,0 + b 0, b,), t 0, = b 0,a 2 = 4 (b 0,0 + b,0 b 0, b,), t, = b,a a 2 = 4 (b 0,0 b,0 b 0, + b,), b 0,0 = b 0,0 + b,0a + b 0,a 2 + b,a a 2, b,0 = b 0,0 b,0a + b 0,a 2 b,a a 2, b 0, = b 0,0 + b,0a b 0,a 2 b,a a 2, b, = b 0,0 b,0a b 0,a 2 + b,a a 2. In these examples, the truth of the results s obvous; and the general demonstraton follows easly from the propertes of the roots of unty. [5.] We have htherto made no use of the assumed rreducblty of the rratonal functon b (m). But takng now ths property nto account, we soon perceve that the component radcals a (k), whch enter nto the composton of ths rreducble functon, must not be subject to, nor even compatble wth, any equatons or equaton of condton whatever, except only the equatons of defnton, whch determne those radcals a (k), by determnng ther prme powers a (k)α(k). For the exstence or possblty of any such equaton of condton n conjuncton wth those equatons of defnton, would enable us to express at least one of the above mentoned radcals as a ratonal functon of others of the same system, and of orders not hgher than ts own, or even, perhaps, as a ratonal functon of the orgnal varables a,... a n, though multpled n general by a root of a numercal equaton; and therefore would enable us to dmnsh the number of extractons of prme roots of functons, whch would be nconsstent wth the rreducblty supposed. If fact, f any such equaton of condton, nvolvng any radcal or radcals of order k, but none of any hgher order, were compatble wth the equatons of defnton; then, by some obvous preparatons, such as brngng the equaton of condton to the form of zero equated to some fnte polynomal functon of some radcal a (k) of the k th order; and rejectng, by the methods of equal roots and of the greatest common measure, all factors of ths polynome, except those whch are unequal among themselves, and are ncluded among the factors of that other polynome whch s equated to zero n the correspondng form of the equaton of defnton of the radcal a (k) ; we should fnd that ths last equaton of defnton a (k)α(k) f (k ) = 0 0

13 must be dvsble, ether dentcally, or at least for some sutable system of values of the remanng radcals, by an equaton of condton of the form a (k)g + g (k) a(k)g + + g (k) g a(k) + g (k) g = 0; g beng less than α (k), and the coeffcents g (k),... g(k) g beng functons of orders not hgher than k, and not nvolvng the radcal a (k). Now, f we were to suppose that, for any system of values of the remanng radcals, the coeffcents g (k),... should all be = 0, or ndeed f even the last of those coeffcents should thus vansh, we should then have a new equaton of condton, namely the followng: f (k ) = 0, whch would be oblged to be compatble wth the equatons of defnton of the remanng radcals, and would therefore ether conduct at last, by a repetton of the same analyss, to a radcal essentally vanshng, and consequently superfluous, among those whch have been supposed to enter nto the composton of the functon b (m) ; or else would brng us back to the dvsblty of an equaton of defnton by an equaton of condton, of the form just now assgned, and wth coeffcents g (k),... g(k) g whch would not all be = 0. But for ths purpose t would be necessary that a relaton, or system of relatons, should exst (or at least should be compatble wth the remanng equatons of defnton,) of the form g (k) g e = ν e a (k)e, e beng less than α (k), and ν e beng dfferent from zero, and beng a root of a numercal equaton; and because α (k) s prme, we could fnd nteger numbers λ and µ, whch would satsfy the condton λα (k) µe = ; so that, fnally, we should have an expresson for the radcal a (k), as a ratonal functon of others of the same system, and of orders not hgher than ts own, though multpled n general (as was above announced) by a root of a numercal equaton; namely the followng expresson: a (k) = ν e µ g (k) µ g e f (k )λ. And f we should suppose ths last equaton to be not dentcally true, but only to hold good for some systems of values of the remanng radcals, of orders not hgher than k, we should stll obtan, at least, an equaton of condton between those remanng radcals, by rasng the expresson just found for a (k) to the power α (k) f (k ) (ν µ e g (k) µ g e ; namely the followng equaton of condton, f (k )λ ) α(k) = 0, whch mght then be treated lke the former, tll at last an expresson should be obtaned, of the knd above announced, for at least one of the remanng radcals. In every case, therefore,

14 we should be conducted to a dmnuton of the number of prme roots of varables n the expresson of the functon b (m), whch consequently would not be rreducble. For example, f an rratonal functon of the m th order contan any radcal a (m) of the cubc form, ts exponent α (m) of defnton beng of the form of the cubc form, ts exponent α (m) α (m)3 = f (m ) (a (m ),... a (m ),... a n (m ),... a n ); beng = 3, and ts equaton f also the other equatons of defnton permt us to suppose that ths radcal may be equal to some ratonal functon of the rest, so that an equaton of the form a (m) + g (m) = 0, (n whch the functon g (m) does not contan the radcal a (m),) s compatble wth the equaton of defnton a (m)3 f (m ) = 0; then, from the forms of these two last mentoned equatons, the latter must be dvsble by the former, at least for some sutable system of values of the remanng radcals: and therefore the followng relaton, whch does not nvolve the radcal a (m), namely, f (m ) + g (m)3 = 0, must be ether dentcally true, n whch case we may substtute for the radcal a (m), n the proposed functon of the m th order, the expresson a (m) = 3. g (m) ; or at least t must be true as an equaton of condton between the remanng radcals, and lable as such to a smlar treatment, conductng to an analogous result. A more smple and specfc example s suppled by the followng functon of the second order, x = a ( ) ( ) c + c 2 c c c 2 c3 2, whch s not uncommonly proposed as an expresson for a root x of the general cubc equaton x 3 + a x 2 + a 2 x + a 3 = 0, c and c 2 beng certan ratonal functons of a, a 2, a 3, whch were assgned n a former artcle, and whch are such that the cubc equaton may be thus wrtten: ( x + a 3 ) 3 3c2 ( x + a 3 2 ) 2c = 0.

15 Puttng ths functon of the second order under the form n whch the radcals are defned as follows, x = a 3 + a + a 2, a 3 = c + a, a 3 2 = c a, a 2 = c 2 c 3 2, we easly perceve that t s permtted by these defntons to suppose that the radcals a, a 2 are connected so as to satsfy the followng equaton of condton, a a 2 = c 2 ; and even that ths supposton must be made, n order to render the proposed functon of the second order a root of the cubc equaton. But the mere knowledge of the compatblty of the equaton of condton a 2 c 2 a = 0 wth the equaton of defnton a 3 2 (c a ) = 0, s suffcent to enable us to nfer, from the forms of these two equatons, that the latter s dvsble by the former, at least for some sutable system of values of the remanng radcals a and a, consstent wth ther equatons of defnton; and therefore that the followng relaton and the expresson c a ( ) 3 c2 a = 0, a 2 = 3. c 2 are at least consstent wth those equatons. In the present example, the relaton thus arrved at s found to be dentcally true, and consequently the radcals a and a reman ndependent of each other; but for the same reason, the radcal a 2 may be changed to the expresson just now gven; so that the proposed functon of the second order, a x = a 3 + a + a 2, may, by the mere defntons of ts radcals, and even wthout attendng to the cubc equaton whch t was desgned to satsfy, be put under the form x = a 3 + a + 3. c 2 a the number of prme roots of varables beng depressed from three to two; and consequently that proposed functon was not rreducble n the sense whch has been already explaned. 3,,

16 [6.] From the foregong propertes of rratonal and rreducble functons, t follows easly that f any one value of any such functon b (m), correspondng to any one system of values of the radcals on whch t depends, be equal to any one root of any equaton of the form x s + a x s + + a s x + a s = 0, n whch the coeffcents a,... a s are any ratonal functons of the n orgnal quanttes a,... a n ; n such a manner that for some one system of values of the radcals a, &c., the equaton b (m)s + a b (m)s + + a s = 0 s satsfed: then the same equaton must be satsfed, also, for all systems of values of those radcals, consstent wth ther equatons of defnton. It s an mmedate consequence of ths result, that all the values of the functon whch has already been denoted by the symbol b (m) must represent roots of the same equaton of the s th degree; and the γ (m),... γ (m) n (m) same prncples show that all these values of b (m) must be unequal among themselves, and γ (m) therefore must represent so many dfferent roots x, x 2,... of the same equatons x s +&c. = 0, f every ndex or exponent γ (m) be restrcted, as before, to denote ether zero or some postve nteger number less than the correspondng exponent α (m) : for f, wth ths restrcton, any two of the values of b (m) could be supposed equal, an equaton of condton between the γ (m),... radcals a (m), &c. would arse, whch would be nconsstent wth the supposed rreducblty of the functon b (m). For example, havng found that the cubc equaton... x 3 + a x 2 + a 2 x + a 3 = 0 s satsfed by the rratonal and rreducble functon b above assgned, we can nfer that the same equaton s satsfed by all the three values b 0, b, b 2 of the functon b γ ; and that these three values must be all unequal among themselves, so that they must represent some three unequal roots x, x 2, x 3, and consequently all the three roots of the cubc equaton proposed. [7.] Combnng the result of the last artcle wth that whch was before obtaned respectng the solatng of a term of a development, we see that f any root x of any proposed equaton, of any degree s, n whch the s coeffcents a,... a s are stll supposed to be ratonal functons of the n orgnal quanttes a,... a n, can be expressed as an rratonal and rreducble functon b (m) of those orgnal quanttes; and f that functon b (m) be developed under the form above assgned; then every term t (m) β (m),... of ths development may be expressed as a ratonal (and ndeed lnear) functon of some or all of the s roots x, x 2,... x s of the same proposed equaton. For example, when we have found that a root x of the cubc equaton x 3 + a x 2 + a 2 x + a 3 = 0 4

17 can be represented by the rratonal and rreducble functon already mentoned, x = b = b 0 + b a + b 2a 2 = t 0 + t + t 2, (n whch b =,) we can express the separate terms of ths last development as follows, t 0 = b 0 = 3 (x + x 2 + x 3 ), t = b a = 3 (x + ρ 3 x 2 + ρ 2 3 x 3), t 2 = b 2a 2 = 3 (x + ρ 2 3 x 2 + ρ 4 3 x 3); namely, by changng b 0, b, b 2 to x, x 2, x 3 n the expressons found before for t 0, t, t 2. In lke manner, when a root x of the bquadratc equaton s represented by the rratonal functon x 4 + a x 3 + a 2 x 2 + a 3 x + a 4 = 0 x = b = b 0,0 + b,0a + b 0,a 2 + b,a a 2 = t 0,0 + t,0 + t 0, + t,, n whch b,0 = b 0, =, we easly derve, from results obtaned before, (by merely changng b 0,0, b 0,, b,0, b, to x, x 2, x 3, x 4,) the followng expressons for the four separate terms of ths development: t 0,0 = b 0,0 = 4 (x + x 2 + x 3 + x 4 ), t,0 = b,0a = 4 (x + x 2 x 3 x 4 ), t 0, = b 0,a 2 = 4 (x x 2 + x 3 x 4 ), t, = b,a a 2 = 4 (x x 2 x 3 + x 4 ); x, x 2, x 3, x 4 beng some four unequal roots, and therefore all the four roots of the proposed bquadratc equaton. And when that equaton has a root represented n ths other way, whch also has been already ndcated, and n whch b =, x = b IV = a 4 + a + a IV = b 0 + b a IV = t IV 0 + t IV, then each of the two terms of ths development may be separately expressed as follows, t IV 0 = b = 2 (x + x 2 ), t IV = b a IV = 2 (x x 2 ), x and x 2 beng some two unequal roots of the same bquadratc equaton. A stll more smple example s suppled by the quadratc equaton, x 2 + a x + a 2 = 0; 5

18 for when we represent a root x of ths equaton as follows, x = b = a 2 + a = t 0 + t, we have the followng well-known expressons for the two terms t 0, t, as ratonal and lnear functons of the roots x, x 2, t 0 = a = 2 2 (x + x 2 ), t = a = 2 (x x 2 ). In these examples, the radcals of hghest order, namely a n b, a n b, a and a 2 n b, and a IV n b IV, have all had the coeffcents of ther frst powers equal to unty; and consequently have been themselves expressed as ratonal (though unsymmetrc) functons of the roots of that equaton n x, whch the functon b (m) satsfes; namely, a = 2 (x x 2 ), a = 3 (x + ρ 2 3x 2 + ρ 3 x 3 ), a = 4 (x + x 2 x 3 x 4 ), a 2 = 4 (x x 2 + x 3 x 4 ), a IV = 2 (x x 2 ); the frst expresson beng connected wth the general quadratc, the second wth the general cubc, and the three last wth the general bquadratc equaton. We shall soon see that all these results are ncluded n one more general. [8.] To llustrate, by a prelmnary example, the reasonngs to whch we are next to proceed, let t be supposed that any two of the terms t (m) are of the forms β (m),... and t 2,,3,4 = b 2,,3,4a 2 a 2a 3 3 a 4 4, t,,2,3 = b,,2,3a a 2a 2 3 a 3 4, n whch the radcals are defned by equatons such as the followng a 3 = f, a 3 2 = f 2, a 5 3 = f 3, a 5 4 = f 4, ther exponents α, α 2, α 3, α 4 beng respectvely equal to the numbers 3, 3, 5, 5. We shall then have, by rasng the two terms t to sutable powers, and attendng to the equatons of defnton, the followng expressons: t 0 2,,3,4 = b 0 2,,3,4f 6 f 2 3 f 3 6 f 4 8 a 2 a 2; t 0,,2,3 = b 0,,2,3f 3 f 2 3 f 3 4 f 4 6 a a 2; t 6 2,,3,4 = b 6 2,,3,4f 4 f 2 2 f 3 3 f 4 4 a 3 3 a 4 4 ; t 6,,2,3 = b 6,,2,3f 2 f 2 2 f 2 3 f 3 4 a 2 3 a 3 4 ; 6

19 whch gve f we put, for abrdgment, t = c a, t 2 = c 2a 2, t 3 = c 3a 3, t 4 = c 4a 4, t = t 0 2,,3,4t 0,,2,3 ; t 2 = t 0 2,,3,4 t 20,,2,3; t 3 = t 8 2,,3,4t 24,,2,3 ; t 4 = t 2 2,,3,4 t 8,,2,3; c = b 0 2,,3,4b 0,,2,3 f 3 f 2 3 f 2 4 ; c 2 = b 0 2,,3,4 b 20,,2,3f 2 3 f 3 2 f 4 4 ; c 3 = b 8 2,,3,4b 24,,2,3 f 4 f 2 2 f 3; c 4 = b 2 2,,3,4 b 8,,2,3f 2 f 2 2 f 4. And, wth a lttle attenton, t becomes clear that the same sort of process may be appled to the terms t (m) of the development of any rreducble functon β (m),... b(m) ; so that we have, n general, a system of relatons, such as the followng: n whch t (m) of the varous terms t (m) β (m) t (m) = c (m ) a (m) ;... t (m) n (m),... = c (m ) a (m) ; n (m) n (m) s the product of certan powers (wth exponents postve, or negatve, or null) ; and the coeffcent c(m ) s dfferent from zero, but s of an order lower than m. For f any radcal of the order m were supposed to be so nextrcably connected, n every term, wth one or more of the remanng radcals of the same hghest order, that t could not be dsentangled from them by a process of the foregong knd; and that thus the foregong analyss of the functon b (m) should be unable to conduct to separate expressons for those radcals; t would then, recprocally, have been unnecessary to calculate them separately, n effectng the synthess of that functon; whch functon, consequently, would not be rreducble. If, for example, the exponents α (m) and α (m) 2, whch enter nto the equatons of defnton of the radcals a (m) and a (m) 2 should both be = 3, so that those radcals should both be cube-roots of functons of lower orders; and f these two cube-roots should enter only by ther product, so that no analyss of the foregong knd could obtan them otherwse than n connexon, and under the form c (m ) a (m) a (m) 2 ; t would then have been suffcent, n effectng the synthess of b (m), to have calculated only the cube-root of the product a (m)3 a (m)3 2 = f (m ) f (m ) 2 = f (m ), nstead of calculatng separately the cuberoots of ts two factors a (m)3 = f (m ), and a (m)3 2 = f (m ) 2 : the number of extractons of prme roots of varables mght, therefore, have been dmnshed n the calculaton of the functon b (m), whch would be nconsstent wth the rreducblty of that functon. In the cases of the rreducble functons b, b, b, b IV, whch have been above assgned, as representng roots of the general quadratc, cubc, and bquadratc equatons, the theorem of the present artcle s seen at once to hold good; because n these the radcals of hghest order are themselves terms of the developments n queston, the coeffcents of ther frst powers beng already equal to unty. Thus n the development of b, we have a = t ; n b, we have a = t ; n b, we have a = t,0, and a 2 = t 0,; and n b IV, we have a IV = t IV [9.] By rasng to the proper powers the general expressons of the form t (m) = c (m ) a (m), 7.

20 we obtan a system of n (m) equatons of ths other form t (m)α(m) = c (m )α(m) f (m ) = f (m ), beng some new rratonal functon, of an order lower than m; and by combnng the same expressons wth those whch defne the varous terms t (m), the number of whch β (m),... f (m ) terms we shall denote by the symbol t (m), we obtan another system of t (m) equatons, of whch the followng s a type, f we put, for abrdgment, u (m ) β (m),... β (m) n (m) = b (m ) β (m),... β (m) n (m), and u (m ) β (m) b (m ) β (m) β (m),... = t(m) =,... b(m ) β (m). (m),... t(m) β. (m),... c(m ) β t (m) β (m) n (m) n (m), c (m ) β (m) n (m) n (m). In ths manner we obtan n general n (m) + t (m) equatons, n each of whch the product of certan powers, (wth postve, negatve, or null exponents,) of the t (m) terms of the development of the rratonal functon b (m), s equated to some other rratonal functon, f (m ) or b (m ), of an order lower than m. Indeed, t s to be observed, that snce these varous equatons are obtaned by an elmnaton of the n (m) radcals of hghest order, between ther n (m) equatons of defnton and the t (m) expressons for the t (m) terms of the development of b (m), they cannot be equvalent to more than t (m) dstnct relatons. But, among them, they must nvolve explctly all the radcals of lower orders, whch enter nto the composton of the rreducble functon b (m). For f any radcal a (k), of order lower than m, were wantng n all the n (m) + t (m) functons of the forms f (m ) and b (m ) β (m),..., we mght then employ nstead of the old system of radcals a (m),... of the order m, a new and equally numerous system of radcals a (m),... accordng to the followng type, a (m) = t (m) = α(m) f (m ) ; and mght then express all the t (m) terms of b (m), by means of these new radcals, accordng to the formula... a (m)β (m) n (m) t (m) β (m) =,... b (m ) β (m). (m),... a (m)β 8 n (m),

21 whch would not nvolve the radcal a (k) ; so that n ths way the number of extractons of prme roots of varables mght be dmnshed, whch would be nconsstent wth the rreducblty of b (m). The results of the present artcle may be exemplfed n the case of any one of the functons b, b, b, b IV, whch have already been consdered. Thus, n the case of the functon b, whch represents a root of the general cubc equaton, we have t = t, c =, f = f, β β = b β, u β and the n (m) + t (m) = + 3 = 4 followng relatons hold good: = t β t 3 = f, t 0 = b 0, = b, t 2t 2 = b 2;. t β, of whch ndeed the thrd s dentcally true, and the second does not nvolve a, because b 0 = a 3 ; but both the frst and fourth of these relatons nvolve that radcal a, because f = c + a, and b 2 = c a c 2. 2 [0.] Snce each of the t (m) terms of the development of b (n) can be expressed as a ratonal functon of the s roots x,... x s of that equaton of the s th degree whch b (m) s supposed to satsfy; t follows that every ratonal functon of these t (m) terms must be lkewse a ratonal functon of those s roots, and must admt, as such, of some fnte number r of values, correspondng to all possble changes of arrangement of the same s roots amongst themselves. The same term or functon must, for the same reason, be tself a root of an equaton of the r th degree, of whch the coeffcents are symmetrcal functons of the s roots, x,... x s, and therefore are ratonal functons of the s coeffcents a,... a s, and ultmately of the n orgnal quanttes a,... a n ; whle the r other roots of ths new equaton are the r other values of the same functon of x,... x s, correspondng to the changes of arrangement just now mentoned. Hence, every one of the n (m) + t (m) functons t (m)α(m) u (m ), and therefore also every one of the β (m),... n(m) + t (m) functons f (m ) and and b (m ), to β (m),... whch they are respectvely equal, and whch have been shown to contan, among them, all the radcals of orders lower than m, must be a root of some such new equaton, although the degree r wll not n general be the same for all. Treatng these new equatons and functons, and the radcals of the order m, as the equaton x s + &c. = 0, the functon b (m), and the radcals of the order m have already been treated; we obtan a new system of relatons, analogous to those already found, and capable of beng thus denoted: t (m ) t (m )α(m ) u (m ) β (m ) = c (m 2) a (m ) ; = f (m 2) ; =,... b (m 2). β (m ),... And so proceedng, we come at last to a system of the form, t = c a,... t n = c n a n ; 9

22 n whch the coeffcent c s dfferent from zero, and s a ratonal functon of the n orgnal quanttes a,... a n ; whle t s a ratonal functon of the s roots x,... x s of that equaton of the s th degree n x whch t has been suppose that b (m) satsfes. We have therefore the expresson a = t c ; whch enables us to consder every radcal a of the frst order, as a ratonal functon f of the s roots x,... x s, and of the n orgnal quanttes a,... a n : so that we may wrte a = f (x,... x s, a,... a n ). But before arrvng at the last mentoned system of relatons, another system of the form t = c a,... t n = c n a n must have been found, n whch the coeffcent c s dfferent from zero, and s a ratonal functon of a,... a n and of a,..., a n, whle t s a ratonal functon of x,... x s ; we have therefore the expresson a = t c and we see that every radcal of the second order also s equal to a ratonal functon of x,... x s and of a,... a n : so that we may wrte a = f (x,... x s, a,... a n ). And re-ascendng thus, through orders hgher and hgher, we fnd, fnally, by smlar reasonngs, that every one of the n + n + + +n (k) + + n (m) radcals whch enter nto the composton of the rratonal and rreducble functon b (m), such as the radcal a (k), must be expressble as a ratonal functon f (k) of the roots x,... x s, and of the orgnal quanttes a,... a n : so that we have a complete system of expressons, for all these radcals, whch are ncluded n the general formula a (k) = f (k) (x,... x s, a,... a n ). Thus, n the case of the cubc equaton and the functon b, when we have arrved at the relaton t 3 = f, n whch we fnd that the ratonal functon t = 3 (x + ρ 2 3x 2 + ρ 3 x 3 ), and f = c + a, t 3 = 27 (x + ρ 2 3x 2 + ρ 3 x 3 ) 3 20,

23 admts only of two dfferent values, n whatever way the arrangement of the three roots x, x 2, x 3 may be changed; t must therefore be tself a root of a quadratc equaton, n whch the coeffcents are symmetrc functons of those three roots, and consequently ratonal functons of a, a 2, a 3 ; namely, the equaton 0 = (t 3 ) 2 27 {(x + ρ 2 3x 2 + ρ 3 x 3 ) 3 + (x + ρ 2 3x 3 + ρ 3 x 2 ) 3 }(t 3 ) (x + ρ 2 3x 2 + ρ 3 x 3 ) 3 (x + ρ 2 3x 3 + ρ 3 x 2 ) 3 ( ) a = (t 3 ) (2a3 9a a a 3 )(t ) + 3a 2. 9 The same quadratc equaton must therefore be satsfed when we substtute for t 3 the functon x + a to whch t s equal, and n whch a s a square root; t must therefore be satsfed by both values of the functon c ± a, because the radcal a must be subject to no condton except that by whch ts square s determned; therefore, ths radcal a must be equal to the sem-dfference of two unequal roots of the same quadratc equaton; that s, to the sem-dfference of the two values of the ratonal functon t 3 ; whch sem-dfference s tself a ratonal functon of x, x 2, x 3, namely a = 54 {(x + ρ 2 3x 2 + ρ 3 x 3 ) 3 (x + ρ 2 3x 3 + ρ 3 x 2 ) 3 } = 8 (ρ2 3 ρ 3 )(x x 2 )(x x 3 )(x 2 x 3 ) = f (x, x 2, x 3 ). The same concluson would have been obtaned, though n a somewhat less smple way, f we had employed the relaton t 2t 2 = b 2, n whch t 2t 2 = 3(x + ρ 2 3x 3 + ρ 3 x 2 ) (x + ρ 2 3 x 2 + ρ 3 x 3 ) 2, b 2 = c a [.] In general, let p be the number of values whch the ratonal functon f (k) can receve, by alterng n all possble ways the arrangements of the s roots x,... x s, these roots beng stll treated as arbtrary and ndependent quanttes, (so that p s equal ether to the product s, or to some submultple of that product); we shall then have an dentcal equaton of the form f (k)p + d f (k)p c d p f (k) + d p = 0, n whch the coeffcents d,... d p are ratonal functons of a,... a n ; and therefore at least one value of the radcal a (k) must satsfy the equaton a (k)p + d a (k)p + + d p a (k) + d p = 0. But n order to do ths, t s necessary, for reasons already explaned, that all values of the same radcal a (k), obtaned by multplyng tself and all ts subordnate radcals of the same 2.

24 functonal system by any powers of the correspondng roots of unty, should satsfy the same equaton; and therefore that the number q of these values of the radcal a (k) should not exceed the degree p of that equaton, or the number of the values of the ratonal functon f (k). Agan, snce we have denoted by q the number of values of the radcal, we must suppose that t satsfes dentcally an equaton of the form a (k)q + e a (k)q + + e q a (k) + e q = 0, the coeffcents e,... e q beng ratonal functons of a,... a n ; and therefore that at least one value of the functon f (k) satsfes the equaton f (k)q + e. f (k)q + + e q f (k) = 0. Suppose now that the s roots x,... x s of the orgnal equaton n x, x s + a x s + + a s x + a s = 0, are really unconnected by any relaton among themselves, a supposton whch requres that s should not be greater than n, snce a,... a s are ratonal functons of a,... a n ; suppose also that a,... a n can be expressed, recprocally, as ratonal functons of a,... a s, a supposton whch requres, recprocally, that n should not be greater than s, because the orgnal quanttes a,... a n are, n ths whole dscusson, consdered as ndependent of each other. Wth these suppostons, whch nvolve the equalty s = n, we may consder the n quanttes a,... a n, and therefore also the q coeffcents e,... e q, as beng symmetrc functons of the n roots x,... x n of the equaton we may also consder f (k) roots, so that we may wrte and snce the truth of the equaton x n + a x n + + a n x + a n = 0; as beng a ratonal but unsymmetrc functon of the same n arbtrary a k = f (k) (x,... x n ); f (k)q + e f (k)q + + e q = 0 must depend only on the forms of the functons, and not on the values of the quanttes whch t nvolves, (those values beng altogether arbtrary,) we may alter n any manner the arrangement of those n arbtrary quanttes x,... x n, and the equaton must stll hold good. But by such changes of arrangement, the symmetrc coeffcents e,... e q reman unchanged, whle the ratonal but unsymmetrc functon f (k) takes, n successon, all those p values of whch t was before supposed to be capable; thse p unequal values therefore must all be roots of the same equaton of the q th degree, and consequently q must not be less than p. And snce t has been shown that the former of these two last mentoned numbers must not exceed the latter, t follows that they must be equal to each other, so that we have the relaton q = p : 22

25 that s, the radcal a (k) and the ratonal functon f (k) must be exactly coextensve n multplcty of value. For example, when, n consderng the rreducble rratonal expresson b for a root of the general cubc, we are conducted to the relaton assgned n the last artcle, a = f (x, x 2, x 3 ) = 8 (ρ3 3 ρ 3 )(x x 2 )(x x 3 )(x 2 x 3 ); we can then at pleasure nfer, ether that the radcal a must admt (as a radcal) of two and only two values, f we have prevously perceved that the ratonal functon f admts (as a ratonal functon) of two values, and only two, correspondng to changes of arrangement of the three roots x, x 2, x 3, namely, the two followng values, whch dffer by ther sgns, ± 8 (ρ3 3 ρ 3 )(x x 2 )(x x 3 )(x 2 x 3 ); or else we may nfer that the functon f admts thus of two values and two only, for all changes of arrangement of x, x 2, x 3, f we have perceved that the radcal a (as beng gven by ts square, a 2 = f = c 2 c 3 2, whch square s ratonal,) admts, tself, of the two values ±a whch dffer n ther sgns. [2.] The condtons assumed n the last artcle are all fulflled, when we suppose the coeffcents a &c. to concde wth the n orgnal quanttes a, &c., that s, when we return to the equaton orgnally proposed; x n + a x n + + a n x + a n = 0, whch s the general equaton of the n th degree: so that we have, for any radcal a (k), whch enters nto the composton of any rratonal and rreducble functon representng any root of any such equaton, an expresson of the form a (k) = f (k) (x,... x n ); the radcal and the ratonal functon beng coextensve n multplcty of value. We are, therefore, conducted thus to the followng mportant theorem, to whch Abel frst was led, by reasonngs somewhat dfferent from the foregong: namely, that f a root x of the general equaton of any partcular degree n can be expressed as an rreducble rratonal functon b (m) of the n arbtrary coeffcents of that equaton, then every radcal a (k), whch enters nto the composton of that functon b (m), must admt of beng expressed as a ratonal, though unsymmetrc functon f (k) of the n arbtrary roots of the same general equaton; and ths ratonal but unsymmetrc functon f (k) must admt of recevng exactly the same varety of values, through changes of arrangement of the n roots on whch t depends, as that whch the radcal a (k) can receve, through multplcatons of tself and of all ts subordnate functonal radcals by any powers of the correspondng roots of unty. Examples of the truth of ths theorem have already been gven, by antcpaton, n the seventh and tenth artcles of ths Essay; to whch we may add, that the radcals a and a, 23

26 n the expressons gven above for a root of the general bquadratc, admt of beng thus expressed: a = 48 {(x + x 2 x 3 x 4 ) 2 + ρ 2 3(x x 2 + x 3 x 4 ) 2 + ρ 3 (x x 2 x 3 + x 4 ) 2 } = 2 {x x 2 + x 3 x 4 + ρ 2 3(x x 3 + x 2 x 4 ) + ρ 3 (x x 4 + x 2 x 3 )}; a = 3456 {x x 2 + x 3 x 4 + ρ 2 3(x x 3 + x 2 x 4 ) + ρ 3 (x x 4 + x 2 x 3 )} {x x 2 + x 3 x 4 + ρ 2 3(x x 4 + x 2 x 3 ) + ρ 3 (x x 3 + x 2 x 4 )} 3 = 52 (ρ2 3 ρ 3 )(x x 2 )(x x 3 )(x x 4 )(x 2 x 3 )(x 2 x 4 )(x 3 x 4 ). But before we proceed to apply ths theorem to prove, n a manner smlar to that of Abel, the mpossblty of obtanng any fnte expresson, rratonal and rreducble, for a root of the general equaton of the ffth degree, t wll be nstructve to apply t, n a new way, (accordng to the announcement made n the second artcle,) to equatons of lower degrees; so as to draw, from those lower equatons, a class of llustratons qute dfferent from those whch have been heretofore adduced: namely, by showng, à pror, wth the help of the same general theorem, that no new fnte functon, rratonal and rreducble, can be found, essentally dstnct n ts radcals from those whch have long snce been dscovered, for expressng any root of any such lower but general equaton, quadratc, cubc or bquadratc, n terms of the coeffcents of that equaton. [3.] Begnnng then wth the general quadratc, x 2 + a x + a 2 = 0, let us endeavour to nvestgate, à pror, wth the help of the foregong theorem, all possble forms of rratonal and rreducble functons b (m), whch can express a root x of ths quadratc, n terms of the two arbtrary coeffcents a, a 2, so as to satsfy dentcally, or ndependently of the values of those two coeffcents, the equaton b (m)2 + a b (m) + a 2 = 0. The two roots of the proposed quadratc beng denoted by the symbols x and x 2, we know that the two coeffcents a and a 2 are equal to the followng symmetrc functons, a = (x + x 2 ), a 2 = x x 2 ; we cannot therefore suppose ether root to be a ratonal functon b of these coeffcents, because an unsymmetrc functon of two arbtrary quanttes cannot be equal to a symmetrc functon of the same; and consequently we must suppose that the exponent m of the order of the sought functon b (m) s greater than 0. The expresson b (m) for x must therefore nvolve at least one radcal a, whch must tself admt of beng expressed as a ratonal but unsymmetrc functon of the two roots x, x 2, a = f (x, x 2 ), 24

27 and of whch some prme power can be expressed as a ratonal functon of the two coeffcents a, a 2, a α = f(a, a 2 ), the exponent α beng equal to the number of the values f (x, x 2 ), f (x 2, x ), of the unsymmetrc functon f, and consequently beng = 2; so that the radcal a must be a square root, and must have two values dfferng n sgn, whch may be thus expressed: +a = f (x, x 2 ), a = f (x 2, x ). But, n general, whatever ratonal functon may be denoted by f, the quotents f(x, x 2 ) + f(x 2, x ) 2 and f(x, x 2 ) f(x 2, x ) 2(x x 2 ) are some symmetrc functons, a and b; so that we may put generally therefore, snce we have, at present, the functon f must be of the form f(x, x 2 ) = a + b(x x 2 ), f(x 2, x ) = a b(x x 2 ); f (x 2, x ) = f (x, x 2 ), f (x, x 2 ) = b(x x 2 ), the multpler b beng symmetrc. At the same tme, and therefore the functon f s of the form so that the radcal a may be thus expressed, a = b(x x 2 ), f (a, a 2 ) = a 2 = b 2 (x x 2 ) 2 = b 2 (a 2 4a 2 ), a = b 2 (a 2 4a 2), n whch, b s some ratonal functon of the coeffcents a, a 2. No other radcal a 2 of the frst order can enter nto the sought rreducble expresson for x; because the same reasonng would show that any such new radcal ought to be reducble to the form a 2 = c(x x 2 ) = c b a, 25

28 c beng some new symmetrc functon of the roots, and consequently some new ratonal functon of the coeffcents; so that, after calculatng the radcal a, t would be unnecessary to effect any new extracton of prme roots for the purpose of calculatng a 2, whch latter radcal would therefore be superfluous. Nor can any radcal a of hgher order enter, because such radcal would have 2α values, α beng greater than, whle any ratonal functon f, of two arbtrary quanttes x, x 2, can receve only two values, through any changes of ther arrangement. The exponent m, of the order of the sought rreducble functon b (m), must therefore be =, and ths functon tself must be of the form b = b 0 + b a, b 0 and b beng ratonal functons of a, a 2, or symmetrc functons of the two roots x, x 2, whch roots must admt of beng separately expressed as follows: x = b 0 + b a, x 2 = b 0 b a, f any expresson of the sought knd can be found for ether of them. It s, therefore, necessary and suffcent for the exstence of such an expresson, that the two followng quanttes, b 0 = x + x 2 2, b = x x 2 2a, should admt of beng expressed as ratonal functons of a, a 2 ; and ths condton s satsfed, snce the foregong relatons gve b 0 = a 2, b = 2b. We fnd, therefore, as the sought rratonal and rreducble expresson, and as the only possble expresson of that knd, (or at least as one wth whch all others must essentally concde,) for a root x of the general quadratc, the followng: x = b = a 2 + b 2b 2 (a 2 4a 2); b stll denotng any arbtrary ratonal functon of the two arbtrary coeffcents a, a 2, or any numercal constant, (such as the number 2, whch was the value of the quantty b n the formulæ of the precedng artcles,) and the two separate roots x, x 2, beng obtaned by takng separately the two sgns of the radcal. And thus we see à pror, that every method, for calculatng a root x of the general quadratc equaton as a functon of the two coeffcents, by any fnte number of addtons, subtractons, multplcatons, dvsons, elevatons to powers, and extractons of prme radcals, (these last extractons beng supposed to be reduced to the smallest possble number,) must nvolve the extracton of some one square-root of the form a = b 2 (a 2 4a 2), and must not nvolve the extracton of any other radcal. But ths square-root a s not essentally dstnct from that whch s usually assgned for the soluton of the general quadratc: t s therefore mpossble to dscover any new rratonal expresson, fnte and rreducble, for a root of that general quadratc, essentally dstnct from the expressons whch have long been known: and the only possble dfference between the extractons of radcals whch are requred n any two methods of soluton, f nether method requre any superfluous extracton, s that these methods may ntroduce dfferent square factors nto the expressons of that quantty or functon f, of whch, n each, the square root a s to be calculated. 26