MEMOIR ON THE UTILITY OF THE METHOD OF TAKING THE MEAN AMONG

Transcription

1 MEMOIR ON THE UTILITY OF THE METHOD OF TAKING THE MEAN AMONG THE RESULTS OF SEVERAL OBSERVATIONS IN WHICH ONE EXAMINES THE ADVANTAGE OF THIS METHOD BY THE CALCULUS OF PROBABILITIES AND WHERE ONE SOLVES DIFFERENT PROBLEMS RELATED TO THIS MATERIAL Joseph Louis Lagrage Miscellaea Tauriesia T V pp 67 3 Oeuvres de Lagrage Vol (867) pp Whe oe has several observatios of the same pheomeo of which the results are ot etirely i accord oe is certai that these observatios are all or at least i part ot very exact from some source that the error could origiate; ow oe has custom to take the mea amog all the results because i this maer the differet errors themselves beig apportioed equally i all the observatios the error which is able to be foud i the resultig mea becomes also the mea amog all the errors Now although everyoe recogizes the utility of this practice i order to dimiish as much as it is possible the ucertaity which is bor of the imperfectio of the istrumets ad of the ievitable errors of observatios I have thought however that it would be good to examie ad to estimate by calculatio the advatages that oe ca hope to extract from a similar method; this is the object that I have proposed to myself i this Memoir I will begi by supposig that the errors which ca slip ito each observatio are give ad that oe kows also the umber of cases which ca give these errors that is to say the facility of each error; I will suppose ext that oe kows oly the limits betwee which all possible errors must be cotaied with the law of their facility ad I will seek through both of the hypotheses what is the probability that the error of the resultig mea be ull or equal to a give quatity or oly cotaied betwee some give limits I will show at the same time how oe ca determie a posteriori the same law of facility of errors ad what is the probability that through this determiatio oe will ot be deceived by a give quatity: whece I will deduce some quite simple rules Traslated by Richard J Pulskamp Departmet of Mathematics & Computer Sciece Xavier Uiversity Ciciati OH November 4 009

2 for the correctio of istrumets by some repeated verificatios For the remaider I will follow i all these researches the ordiary rule of calculatio of probabilities followig which oe estimates the probability of a evet by the umber of favorable cases divided by the umber of all possible cases The difficulty cosists oly i the eumeratio of these cases; but this eumeratio demads ofte some quite complicated calculatios ad of which oe ca come to the ed oly by some particular artifices: this is that which takes place maily i the material that I am goig to treat PROBLEM I Oe supposes that i each observatio oe ca be deceived by oe uit as much to the greater as to the lesser but that the umber of cases which ca give a exact result is to the umber of cases which ca give a error of oe uit is as a : b; oe demads what is the probability of havig a exact result i takig the mea amog the particular results of a umber of observatios Sice there are a cases which give zero error ad b cases which give ad that is to say b cases which give ad b cases which give error it is clear by the ordiary rules of probabilities that the probability that the error be ull i each a ab particular observatio will be expressed by ; we see therefore what will be the probability that the error be also ull i takig the mea amog observatios It is easy to see that this questio is reduced to this oe: Havig dice of which each have a faces marked with a zero b faces marked with a positive uit ad b faces marked with a egative uit so that the total umber of faces be a b to fid the probability that there is brought zero i castig all these dice at radom Now oe kows by the theory of combiatios that if oe raises the triomial ab(xx ) to the power the coefficiet of the absolute term that is to say the oe where the power of x will be zero will deote the umber of cases or chaces where the sum of the poits marked by all the dice will be equal to zero: therefore amig this coefficiet A oe will have because the umber of all possible combiatios is (a b) A oe will have I say (ab) for the sought probability All is reduced therefore to fidig the coefficiet of A; ow this is to what oe ca attai by several differet ways If oe develops the power [a b(x x )] followig the theorem of Newto oe will have as oe kows a a b(x x ) ( ) a b (x x ) ; ow it is easy to see that the odd powers of x x cotai o term without x ad that i the eve powers there is always a term without x which is the oe i the middle i which the expoets of x ad x are the same Thus the term without x of (x x ) will be the oe of (x x ) 4 will be 43 the oe of (x x ) 6 will

3 be ad thus the others; therefore oe will have i geeral that is to say A =a ( ) a b 43 ( )( )( 3) a 4 b ( )( )( 3)( 4)( 5) a 6 b A =a ( )a b ( )( )( 3) a 4 b 4 ( )( )( 3)( 4)( 5) a 6 b 6 33 It is clear that the triomial a b(x x ) ca be factored ito these two biomials αβx αβx this which gives by compariso of the terms α β = a ad αβ = b; whece oe draws α ± β = a ± b ad from there a b a b α = a b a b β = This put oe will have therefore ( [a b(x x )] = (α βx) α β x [ = α α βx [ α α β x ) ] ( ) α β x ] ( ) α β x whece it is easy to coclude that oe will have [ ( )α A = α (α β) β ] [ ( )( )α 3 β 3 ] 3 3 COROLLARY Let a = b that is to say let there be a equal umber of cases which give 0 or or error; the probability of havig a exact result i a each particular observatio will be ab = 3 ad that of havig a exact result i takig the middle term amog the results of observatios will be followig the first A formula [i dividig the upper ad the lower of the fractio (ab) by a ] ( ) ( )( )( 3) ( ) ( 5) 33 3 Therefore i takig successively equal to 3 oe will have Probability

4 Oe sees by this table that the probability that the error be ull dimiishes i measure as oe take a greater umber of observatios so that if oe would wish to estimate the advatage that there ca be i takig the mea amog several observatios by the excess of the probability that the error be ull i the resultig mea over the oe that the error be ull also i each particular result oe will fid i the case i questio here that the advatage will always be egative that is to say that it will chage to disadvatage which would proceed likewise by icreasig the more observatios there would be; whece it seems that oe could coclude that i this case it would be more desirable to take a uique observatio tha to take the mea amog several observatios; but there is a essetial cosideratio to make o this matter from which it results that it is always more advatageous i practice to multiply observatios as ofte as oe ca: it is this that we will discuss more below 3 COROLLARY II Let ow a = b so that the umber of cases which give a exact result is equal to the umber of those which ca give a error of or I this case it will be worth more to avail oeself of the secod formula because oe will have α = b β = b so that because a b = 4b oe will have o dividig A the upper ad lower of the fractio (ab) by b [ ( ) ] [ ] ( )( ) 3 4 for the probability that the error be ull i takig the mea amog observatios Therefore makig successively equal to 3 oe will have the followig results 3 4 Probability whece oe sees that the probability dimiishes i measure as icreases as i the case of the precedig corollary 4 COROLLARY III Let b = a i such a way that the umber of cases which ca give a error of oe uit as may by more as by less is double of the oe where oe would have a exact result oe will have here for the probability that the error be ull i takig the mea amog several observatios 4( ) 6( )( )( 3) 6( ) ( 5) 33 5 Therefore makig successively equal to 3 oe will have 3 4 Probability Thus for two observatios the advatage will be of = 4 5 for three it will be of = 0 for four equal to 5 5 = 4 5 etc; whece it appears that the greater advatage takes place i takig the mea betwee two observatios oly 5 REMARK I order to facilitate further the solutio of the precedig Problem it is good to seek the law that follows the terms of the series which represets 4

5 the probabilities which correspod to 3 observatios; ow if oe takes the fractio z[ab(xx )] ad if oe develops it ito a series accordig to the powers of z oe will have as oe kows z[a b(x x )] z [a b(x x )] such that i this series the coefficiet of z will the be the th power of a b(x x ); therefore if oe ames A A A the values of A which correspod to = 3 that is to say the terms without x of the powers of a b(x x ) [a b(x x )] it is clear that the series A z A z A z 3 will be equal to the sum of the terms without x i the fractio z[ab(xx )] developed accordig to the powers of x ad of x ; such that if oe represets by ( Z Z x ) Z (x x ) x the series which results from the developmet of this fractio accordig to the powers of x ad of x (because it is easy to see that the series i questio must have ecessarily this form) oe will have Z = A z A z ; thus kowig the fuctio Z there will be o more tha to reduce it to a series accordig to the powers of z i order to have the quatities A A A A For this I reduce first the triomial az bz(x x ) to (p qx)(p qx ) that which gives me ext I reduce the fractio p q = az ad pq = bz; (p qx)(p qx ) to α β p qx β p qx ad I fid ow ad likewise α = q p β = p p q ; p qx = p qx p q x p 3 p qx = p q p x q p 3 x ; 5

6 therefore oe will have Z = α β p Z = βq p Z = βq p 3 Z = βq3 p 4 ; therefore Z = q p q p = p q = (p q)(p q) ; but sice p q = az ad pq = bz oe will have p q = az bz p q = az bz; therefore therefore fially (p q)(p q) = ( az) 4b z ; Z = az (a 4b )z = A z A z A z 3 such that oe will have for the kow fuctios A = a A = 3aA 4b a A = 5aA (4b a )A 3 A iv = 7aA 3(4b a )A We deote by P P P the probabilities that the error be ull i takig the mea amog 3 observatios ad oe will have whece P = A a b P = 4 A (a b) P = A (a b) 3 A = (a b)p A = (a b) P A = (a b) 3 P ; therefore substitutig these values ito the precedig formulas ad makig for greater 6

7 simplicity b a = r oe will have P = r P = 3P r ( r) P = 5P (r )P 3( r) P iv = 7P 3(r )P 4( r) P v = 9P iv 4(r )P 5( r) 6 REMARK II If oe makes r = oe will have the case of Corollary II where a = b ad oe will fid P = P = 3 4 P = ad i geeral P () 35 ( ) = 46 Thece oe sees that the probability always dimiishes i measure as icreases this which we have already observed i the Corollary cited; so that i takig = the probability will become ifiitely small or ull; ideed by the quadrature of Wallis oe has (π beig the arc of 80 degrees) that is to say i takig = π = π = ( )( )( ) ; therefore multiplyig by ad takig the square root oe will have 46 π = 35 ( ) ; therefore whe = oe will have P () = π = 0 It is good to remark that sice we have foud i the cited corollary for the probability P the expressio [ ] [ ] () ( )( ) 3 4 7

8 oe will have i comparig this expressio with the precedig the equatio [ ] [ ] ( ) ( )( ) = 3 35 ( ) 3 which is the much more remarkable that it does ot appear easy to demostrate a priori 7 REMARK III By the formulas of Remark I oe will have i geeral P () = ( )P ( ) ( )(r )P ( ) (r ) P () = ( )P () (r )P ( ) ( )(r ) P () = ( 3)P () ( )(r )P () ( )(r ) where the expoets etc of P do ot deote the powers but oly the positio of the rak Now if is a very great umber it is clear that the fractios 3 etc will be very early equal to ad that the fractios will be also very early equal to ; so that oe will have by this hypothesis P () = P ( ) (r )P ( ) r P () = P () (r )P ( ) r whece oe sees that the quatities P () P () etc form a recurret series of which the deomiator of the geeratig fractio will be x x r r r ; thus oe will have i geeral [ P (s) ] s [ 4r = A 3 ] s 4r B 3 (r ) ( r) ad i order to determie the coefficiets A ad B oe will suppose that the terms P () ad P () are kow this which will give P () = A B ad P () = A 4r 3 (r ) B 4r 3 ( r) 8

9 whece whece [ P P (s) () = A = ( r)p () ( 4r 3)P () 4r 3 B = ( 4r 3)P () ( r)p () 4r 3 ( r)p () P () 4r 3 ] [ 4r 3 ( r) [ P () ( r)p () P () ] [ 4r 3 4r 3 ( r) ] s ] s ad this formula will be so much the more exact as oe takes the umber more great Thus after havig calculated the terms P () ad P () either by the formulas of No or by those of Remark I oe could fid very early all the followig terms by the precedig formula To the remaider it is easy to see by this formula that the probability will be ull at ifiity that is to say whe s = ; ideed it is clear that whatever be r provided that it be a positive umber the quatities ± 4r 3 (r) will be always smaller tha ; because we suppose if it is possible ± 4r 3 (r) > oe will have therefore amely ad amely 4r ± 4r 3 > 4( r r ) ± 4r 3 > 3 4r 4r 3 > 6r 4r 9 0 > r 4r this which caot be; therefore by makig s = the quatities [ ] s [ 4r 3 ] s 4r ad 3 ( r) ( r) will become ull ad cosequetly P () also 8 SCHOLIUM Let ρ be the result that each observatio must give if it were exact: sice oe supposes that oe ca be deceived by oe uit as much to the greater as to the less oe will have i each observatio oe of these three results: ρ ρ ρ ; therefore if oe has two observatios ad if oe takes the mea betwee their results that is to say the half-sum of these results oe will have oe of these five results ρ ρ ρ ρ ρ 9

10 amely ρ ρ ρ ρ ρ ; thus i this case the error ca be or as much to the greater as to the less; oe will see similarly that i takig the mea amog three observatios the error could be or 3 or 3 as much to the greater as to the less ad thus so forth Thus although the probability that the error be ull ca be smaller whe oe takes the mea result of several observatios tha whe oe takes the result of each observatio i particular however if oe seeks the probability that the error ot surpass or 3 oe will fid that this probability will be greater i the first case tha i the secod Ideed i the first case there are o other favorable cases tha those where the error is absolutely ull; but i the secod the favorable cases are ot oly those where the error is ull but also those where the error is or 3 ad it is by this cosideratio that it is always more advatageous to take the mea amog the results of several observatios tha to cotai oeself to the result of each observatio i particular We are goig to examie the questio uder this poit of view i the followig Problem PROBLEM II 9 The same thigs beig supposed as i the precedig Problem to fid the probability that i takig the mea amog the results of observatios the error will ot surpass the fractio m m beig < I takig the mea amog the results of observatios it is clear that the error ca be: either 0 or ± or ± or ± 3 util ± amely ± Thus the probability that the error ot be greater tha ± m will be the sum of the probabilities that the error will be ull or ± or ± util ± m We see therefore first what it is the probability that the error will be ± µ I revertig this questio to the dice as we have practiced i Problem I it is clear that it is reduced to fidig the probability of brigig µ or µ poits with dice of which each has a faces marked 0 b faces marked ad b faces marked For this there is oly to raise the triomial a b(x x ) to the power ad the coefficiet of x µ will deote the umber of cases where the sum of the poits of all the dice will be µ similarly that the oe of x µ will deote the umber of cases where the sum of the poits will be µ; thus the sum of these two coefficiets divided by (a b) which is the umber of all cases will give the sought probability Now oe has [a b(x x )] = a a b(x x ) ( ) a b (x x ) 0

11 ad moreover (x x ) = (x x ) (x x ) 3 = (x 3 x 3 ) 3(x x ) (x x ) 4 = (x 4 x 4 ) 4(x x ) 43 (x x ) 5 = (x 5 x 5 ) 5(x 3 x 3 ) 54 (x x ) Therefore if oe supposes [a b(x x )] = A B(x x ) C(x x ) D(x 3 x 3 ) oe will have A =a ( ) a b 43 ( )( )( 3) a 4 b ( ) ( 5) a 6 b B =a b 3 ( )( ) a 3 b ( )( )( 3)( 4) a 5 b ( ) ( 6) a 7 b ( ) C = a b 4 ( )( )( 3) a 4 b ( ) ( 5) a 6 b ( ) ( 7) a 8 b Therefore if oe calls M the term of the series A B C of which the idex will

12 be µ it is easy to see that oe will have ( ) ( µ ) M = a µ b µ µ µ ( ) ( µ ) a µ b µ (µ ) (µ 4)(µ 3) ( ) ( µ 3) a µ 4 b µ4 (µ 4) Now this term M is the coefficiet of the powers of x µ ad x µ such that oe will M have (ab) for the probability that the error be ± µ Thus the probability that the error will ot surpass ± µ will be represeted by the series A B C D M (a b) I order to facilitate the search for the values of A B C it is good to show how these quatities deped o oe aother; for this oe will take the equatio [a b(x x )] = A B(x x ) C(x x ) D(x 3 x 3 ) ad takig the logarithmic differetial oe will have after havig divided by dx x b(x x ) a b(x x ) = B(x x ) C(x x ) A B(x x ) C(x x ) ; therefore cross multiplyig there will happe ba(x x ) bb(x x ) bc(x 3 x 3 x x ) bd(x 4 x 4 x x ) =ab(x x ) ac(x x ) 3aD(x 3 x 3 ) bb(x x ) bc(x 3 x 3 x x ) 3bD(x 4 x 4 x x ) ; so that i comparig the terms oe will have b(a C) = ab bc b(b D) = ac b(b 3D) b(c E) = 3aD b(c 4E)

13 whece by makig for greater simplicity a b = K oe will have A KB C = ( )B KC D = 3 ( )C 3KD E = 4 Thus by kowig the first two terms A ad B oe could fid successively all the others 0 COROLLARY Suppose as i No a = b so that oe has K = ad makig successively equal to 3 ad a = this which is permissible oe will fid the followig values A B C D E F G Thece oe will form the followig table of probabilities: VALUES of the umber of observatios PROBABILITIES THAT THE ERROR WILL NOT SURPASS THE FRACTIONS ± 0 ± ± ± 3 ± 4 ± Oe sees by this table that i takig the mea betwee two observatios the probability that the error be ull will be 3 9 = 3 ad the oe that the error will ot surpass so much to the greater as to the lesser will be 7 9 ; ow i each particular observatio there is a 3 probability that the error will be 0 ad sice by hypothesis the error ca be oly 0 or ± it is clear that the probability that the error will ot surpass will be the same 3 Thus although the probability that the error will be

14 ull is the same either that oe take the mea result betwee two observatios or that oe takes the particular result of a uique observatio however the probability that the error will ot surpass will be greater i the first case tha i the secod these two probabilities beig as 7 9 : 3 that is to say i the ratio of 7 : 3 Likewise i takig the mea amog three observatios oe will have 7 7 for the probability that the error will be ull 9 7 for the probability that the error will ot be greater tha ± 5 3 ad 7 for the oe that the error will ot be greater tha ± 3 ; but i each particular observatio the probability that the error be ull is 3 ad the oe that the error ot surpass 3 or 3 is the same 3 because by hypothesis the error ca oly be ull or ±; therefore the probability that the error be ull will be i truth greater i the particular result of oe uique observatio tha i the mea result of three observatios ad this i the ratio of 9 : 7; but i retur the probability that the error will ot surpass ± 3 will be greater i the secod case tha i the first by ratio of 9 : 9 ad the oe that the error will ot surpass ± 3 it will be agai more this probability beig i the secod case greater tha i the first by ratio of 5 : 9 Here is therefore i what cosists pricipally the advatage that there is i takig the mea amog the results of several observatios I order to reder the thig agai more sesible we are goig to seek the probabilities that the error will ot surpass the fractio i supposig successively equal to 3 that is to say for a uique observatio for two for three ad we will have Probabilities or else by reducig to the same deomiator Probabilities Oe sees thece that the probability that the error will ot surpass proceeds by icreasig i measure as oe takes a greater umber of observatios but with this differece that the probability is greater for two observatios tha for three for four tha for five ad i geeral for ay eve umber tha for the odd umber which follows it immediately; so that i the hypothesis i questio is it is more advatageous to take the mea oly amog some eve umber of observatios REMARK We have see i No 5 that if oe develops the fractio z [ a b ( x x)] i a series of this form ( Z Z x ) Z (x x ) x Z Z Z beig some fuctios of z oe will have Z = p q p ad q beig such that Z = βq p = q p Z Z = βq p 3 = q p Z p q = az ad pq = bz 4

15 this which gives ad thece so that by makig for greater simplicity oe will have p q = az (a 4b )z q p = az az (a 4b )z ; bz ζ = az (a 4b )z Z = ζ Z = az ζ bz ζ Z = ( az ζ bz ) ζ ad i geeral ( ) µ az ζ Z (µ) = bz ζ Now if oe develops this quatity ito a series of ratioal ad iteger powers of z oe will see easily by that which we have said above that the coefficiet of ay oe power as z will deote the umber of cases where the sum of the errors of observatios could be µ or µ so that the double of this coefficiet will express the umber of all cases where the mea error will be ± µ Thece it is easy to coclude that the quatity az ζ bz ( az ζ bz ) ( ) µ az ζ bz ζ beig regarded as a fuctio of z ad developed accordig to the powers of this variable will give a series of such ature that the coefficiet of ay power z will express correctly the umber of cases where the mea error could be cotaied i these limits µ µ ; such that this coefficiet beig divided by the total umber of cases (ab) oe will have the value of the probability that the mea error will ot surpass the fractio µ either to the greater or to the lesser Now the quatity i questio beig othig other tha a geometric series it ca be put uder this more simple form ( ζ ( ) µ az ζ bz az ζ bz ) ζ Thus the etire difficulty will cosist i reducig this same quatity to a ifiite series which proceeds accordig to the powers of z I order to come more easily to the ed oe will suppose it equal to a idetermiate y ad oe will have a equatio betwee y ad z that oe could deliver by some differetiatios as much from the power µ as from the irratioality of ζ; by this meas oe will have a differetial equatio of secod degree betwee y ad z ad there will be o more tha to suppose 5

16 y = Az Bz ad to determie the coefficiets A B by the compariso of the terms Moreover as this calculatio is a little log we will cotet ourselves to idicate here i order to put o the way the oes who will wish to push this theory further SCHOLIUM We have supposed i the two precedig Problems that there were a equal umber of cases i order to have a positive error ad i order to have a egative; if this were ot thus ad if the umber of cases of error which would give 0 ad were a b ad c ow oe would solve the Problems with the same facility by cosiderig the triomial a bx cx i the place of a b(x x ) i order to have the umber of cases where oe would have a give mea error ad i takig ext (a b c) i order to have the total umber of cases i the place of (a b) Oe could likewise without makig a ew calculatio adapt to this case here the formulas that we have already foud; because if i the triomial a bx c x oe puts x c b i the place of x it will become a bc ( x x) ; thus there would be oly to put i the triomial a b ( x x) of the precedig Problems bc i the place of b ad ext b x c i the place of x Of the rest we are goig to treat this case i a much more geeral maer i the followig Problem PROBLEM III 3 Supposig that each observatio be subject to a error of oe uit to the lesser ad to a error of r uits to the greater ad that the umber of cases of error which ca give 0 r are respectively a b c; oe demads what is the probability that the mea error of several observatios will be cotaied withi some give limits Let be the umber of observatios of which oe wishes to take the mea; oe will form the th power of the triomial a b x cxr ad the coefficiet of ay power x µ will deote the umber of cases where the sum of the errors will be µ ad cosequetly where the mea error will be µ We will cosider therefore the quatity ( a b ) x cxr which is reduced to ad oe will have as oe kows [b x(a cx r )] x [b x(a cx r )] = b b x(a cx r ) ( ) b x (a cx r ) 6

17 whece it is easy to see that the coefficiet of ay power x s will be ( ) ( s ) b s a s 3 s ( ) ( s r) s r b sr a s r c 3 (s r) ( ) ( s r) (s r)(s r ) b sr a s r c 3 (s r) this series beig cotiued util that which oe attais to some egative terms; therefore this coefficiet will be the oe of the power x s i the quatity ( a b x cxr) s ; therefore if oe desigates i geeral by (µ) the coefficiet of the power x µ i this last quatity oe will have ( ) ( µ) (µ) = b µ a µ 3 (µ ) ( ) (r µ) 3 (µ r ) br µ a µ r c ( ) (r µ) 3 (µ r ) br µ a µ r c where it will be ecessary to omit the terms which would cotai some egative powers of a or b Therefore sice for observatios the sum of all the cases is (a b c) oe will have for the probability that the mea error be µ the quatity (µ) (abc) ; ad thece the probability that the mea error will be cotaied betwee these limits p q will be expressed by the series ( p ) ( ) (0) () (q ) (a b c) PROBLEM IV 4 Supposig all as i the precedig Problem oe demads what is the mea error for which the probability is the greatest We have see that the probability that the mea error be µ is (µ) (abc) (µ) beig the coefficiet of the power x µ of the triomial ( a b x cxr) ; thus the questio is oly to kow what is the term of the the power of a b x cxr which will have the greatest coefficiet; for this it is clear that there is oly to seek the greatest term of the triomial abc raised to the power ; because supposig that this term be πa α b β c γ α β γ beig the expoets of a b c of which the sum must be equal to ad π the 7

18 coefficiet of this term there will be oly to put b x i the place of b ad cxr i the place of c ad oe will have πa α b β c γ x βrγ for the sought term of the th power of the triomial a b x cxr ; thus oe will make β rγ = µ ad oe will have rγ β for the mea error of which the probability will be greatest Now by the rules of combiatios oe kows that the coefficiet π of the term πa α b β c γ must be 3 3 α 3 β 3 γ ; we deote this term by M so that oe has 3 a α b β c γ 3 α 3 β 3 γ = M ad it will be ecessary that i varyig the expoets α β γ the value of M dimiishes; we make therefore α vary by oe uit so that α becomes α ad as α β γ = it will be ecessary that β or γ dimiish at the same time by oe uit; ow it is easy to see that if i the value of M oe puts α for α ad β for β this value will become β am α b therefore ad cosequetly β am α b < M β a α b < ; reciprocally if oe augmets β by oe uit ad if oe dimiishes α also by oe uit oe will fid the coditio α b β a < ; thus it will be ecessary that oe have at the same time α β < a b ad α > a β b Now this is that which will take place if α β = a b Oe will fid i the same maer α γ = a c ; so that i takig a idetermiate coefficiet p oe will have i the case of the maximum α = pa β = pb γ = pc; 8

19 but α β γ = ; therefore p = abc ; therefore fially If the quatities α = a abc α = a a b c β = b a b c γ = c a b c b abc c abc are whole umbers oe will have exactly a a b c β = b a b c γ = c a b c as we happe to fid it; but if these quatities are ot some whole umbers the it will be ecessary to take for α β γ the whole umbers which will be the earest Oe ca take however for greater simplicity these same quatities for the values of α β γ because the error if there is could ever be oly quite small; i this maer we will have for the mea error what has the greatest probability the expressio rγ β = rc b a b c 5 COROLLARY Thece it follows that oe ca always regard the quatity rc b abc as the error of the mea result ad that also oe ca take the same quatity for the correctio of this result Whe r = ad c = b as i the hypothesis of Problem I the correctio of the mea result becomes ull; it would be also it if oe would have b = rc; but i all the other cases it will be so much greater as rc will differ ay more from b PROBLEM V 6 Oe supposes that each observatio be subject to some errors ay whatsoever give ad that oe kows at the same time the umber of cases where each error ca take place; oe demads the correctio that it will be ecessary to make to the mea result of several observatios Let p q r s be the errors to which each observatio is subject ad a b c d the cases which ca give these errors amely a the umber of cases which could give the error p b the umber of cases which could give the error q ad thus the others; it is clear by that which we have demostrated i the precedig Problems that if oe raises the polyomial ax p bx q cx r to the power ad if oe deotes by M the coefficiet of the power x µ oe will have M (a b c ) for the probability that the error of the mea result of observatios be µ Now oe kows by the theory of combiatios that the coefficiet M will be of this form 34 a α b β c γ 3 α 3 β 3 γ 9

20 where the expoets α β γ must be such that α β γ = ad αp βq γr = µ Moreover it is easy to demostrate by a similar method to that of the precedig Problem that the coefficiet M will be the greatest whe oe will have a α = a b c b β = a b c c γ = a b c ; whece it follows that the mea error for which the probability will be the greatest will be expressed by µ ap bq cr = a b c Thus this quatity will represet the correctio that it will be ecessary to make to the mea result of several observatios 7 COROLLARY I If oe regards the quatities a b c as some weights applied to a idefiite lie at some distaces equal to p q r from a fixed poit take o this lie ad if oe seeks the ceter of gravity of these weights the distace of this ceter to the fixed poit will be the correctio that it will be ecessary to make to the mea result of several observatios; this follows evidetly from the formula that we have foud above for the value of this correctio 8 COROLLARY II Therefore if oe supposes that each observatio be subject to all possible errors which ca be cotaied betwee the give limits ad if oe kows the curve of the facility of errors i which the abscissas beig supposed to represet the errors the ordiates represet the facilities of these errors there will be oly to seek the ceter of gravity of the total area of this curve ad the abscissa correspodig to this ceter will express the correctio of the mea result Thece oe sees that if the curve i questio is equal is similar o all sides of the ordiate which passes through the origi of the abscissas so that this ordiate is a diameter of the curve i questio the the correctio will be ull the ceter of gravity fallig ecessarily o the diameter This case takes place all the time that the errors ca be equally positive ad egative PROBLEM VI 9 I suppose that oe has verified a arbitrary istrumet ad that havig repeated several times the same verificatio oe has foud differet errors of which each is foud repeated a certai umber of times; oe demads what is the error that it will be ecessary for the correctio of the istrumet Let p q r be the foud errors ad let α β γ be the umbers which mark how may times each error is foud repeated i makig verificatios; suppose that the 0

21 umber of cases which ca give the error p or q or r are desigated respectively by a b c ; let oe raise the polyomial to the power ad let ax p bx q cx r N(ax p ) α (bx q ) β (cx r ) γ be ay term of this polyomial: the coefficiet Na α b β c γ of the power pα qβ rγ of x divided by (a b c ) will deote the probability that the errors p q r are foud combied together i a way that p is repeated α times q β times r γ times ad thus the others Thus this probability will be the greatest i the combiatio where the value of Na α b β c γ will be the greatest; but oe has N = 34 3 α 3 β 3 γ as we have see already i the precedig Problem; therefore by the same Problem the greatest value of Na α b β c γ will take place whe a α = a b c b β = a b c c γ = a b c ; equatios by which oe could determie the ukows a b c ; ad oe will have by makig a b c = s a = sα b = sβ c = sγ Now we have demostrated i the Problem cited that the correctio that it is ecessary to make to the mea result of a arbitrary umber of observatios is expressed by ap bq cr ; a b c therefore puttig i this expressio the values of a b c that we happe to fid the correctio i questio will become αp βq γr that is to say equal to the mea error amog all the particular errors that the verificatios have give 0 COROLLARY If oe would wish to keep cout also at least i a approximate maer of itermediate errors to which the istrumet could be subject there would be oly to take o a idefiite straight lie some abscissas proportioal to the

22 foud errors p q r as i No 7; ad havig applied there some ordiates proportioal to the quatities a b c oe would pass through the extremities p q r a parabolic lie; oe would seek ext the ceter of gravity of the area of all the curve ad the perpedicular dropped from this ceter oto the axis would cut there a abscissa which would be the correctio to the istrumet Oe sees thece how oe ca kow a posteriori the law of facility of each of the errors to which a istrumet ca be subject REMARK I Oe has foud above that the greatest probability takes place whe a = sα b = sβ c = sγ so that the values of a b c are the most probable that oe ca suppose If oe would wish to kow moreover what is the probability that these same values will ot depart from the truth of ay quatity ± rs there will be oly to put i the geeral expressio of the probability (precedig Problem) Na α b β c γ s i place of a b c the quatities s(αx) ad makig successively x y z equal to ± ± ±3 ±r so that meawhile if oe has always xyz = 0 because (by hypothesis) abc = s ad αβγ = oe will have as may particular probabilities of which the sum will be the sought probability Let P be the probability that oe has s(βy) s(γz) a = sα b = sβ c = sγ ; puttig these values i the precedig expressio oe will have P = 3 µ α α Let moreover Q be the probability that oe has a = s(α x) b = β β 3 α 3 β s(β y) c = s(γ z) oe will have the value of Q i puttig these values ito the same expressio ad it is easy to see that oe will have Q = P Therefore if oe has i geeral V = ( x ) ( α y ) β ( z ) γ α β γ ( x ) ( α y ) β ( z ) γ α β γ

23 ad if V deotes the sum of all the particular values of V i varyig x y z from 0 to r ad havig care that oe has always x y z = 0 the sought probability will be equal to P V Sice it is ot easy to fid the itegral V maily whe there are more tha two variables oe could be cotet to have a approximate method; for this there would be oly to take a mea value of V ad to multiply it by the umber of all the particular values of V which must eter ito the itegral V ad the difficulty will cosist oly i fidig this umber Now if oe desigates by m the umber of the quatities α β γ it is easy to imagie that the umber i questio will be oe other tha the coefficiet of u 0 that is to say the well kow term of the series which represets the m power of the polyomial u r u r u u u r u r Let oe deote this term by T ad oe will have as we will demostrate below (mr )(mr )(mr 3) (mr m ) T = 3 (m ) [(m )r][(m )r ][(m )r 3] [(m )r m ] m 3 (m ) m(m ) [(m 4)r ][(m 4)r][(m 4)r ] [(m 4)r m 3] 3 (m ) by cotiuig this series oly util whe some oe of the factors mr (m )r (m 4)r becomes egative Therefore if W is the mea value of V oe will have for the approximate value of V the quatity V W ad the sought probability will be very early equal to P T W If istead of takig for W the mea value of V oe takes the least it is clear that T W will be ecessarily less tha the true value of V ad cosequetly the sought probability will be ecessarily greater tha P T W ; thus oe could wager with advatage P T W agaist P T W that by makig a s = α b s = β c s = γ oe will ot be deceived by a quatity greater tha r as much to the greater as to the lesser REMARK II Suppose that is a very great umber ad that cosequetly the umbers α β γ of which the sum is are also very great; i order to fid i this case the values of P ad of V oe will remark: That whe u is a very great umber oe has very early log log log 3 log u = log π log u u π beig the ratio of the periphery of the circle to the radius; whece it follows that oe will have log 3 u u u = log π log u u 3

24 ad cosequetly 3 u πu u u = e u ; therefore because α β γ = oe will have π P = πα πβ πγ If oe takes the logarithm of V oe will have ( log V = α log x ) ( β log y α β ) γ log but ( log x ) = x α α x α therefore because x y z = 0 oe will have very early log V = ( ) x α y β z γ ( z ) γ ad thece Now let there be V = e ( x α y β z γ ) ad oe will have therefore x = ξ y = ψ z = ζ α = A β = B γ = C ξ ψ ζ = 0 ad A B C = ; P = V = e (π) m ABC ( ξ A ψ B ζ C ) Now as the icremet or the differece of the quatities x y z is the differece of the variables ξ ψ ζ will be ad cosequetly ifiitely small; so that if oe calls this differece dθ oe will have P = dθ m π m ABC Therefore ( ξ ) P V = e A ψ B ζ C dθ m π m ABC 4

25 Therefore if oe itegrates the differetial dθ m ( ) e ξ A ψ B ζ C m times i puttig first i the place of ξ its value ψ ζ ad varyig ext successively the variables ψ ζ of the same differetial dθ ad if oe completes the itegral so that the values of ξ ψ ζ exted from ρ util ρ (by makig r = ρ ) oe will have by amig this itegral R the quatity R π m ABC for the probability that the values of a b c will be early exact to ρs Let for example m = so that oe has foud oly two differet errors of which the oe has bee repeated α times ad the other β times i a very great umber of verificatios of the istrumet; i this case there will be oly oe itegratio to make ad the differetial to itegrate will be i puttig ψ i the place of ξ ad makig dθ = dψ dψ e ( A B )ψ which is ot itegrable by ay of the kow methods uless oe reduces to series the expoetial quatity e ( A B )ψ I this maer oe will have the differetial dψ ( Kψ K ψ 4 K3 ψ 6 3 by makig for brevity K = AB AB ; so that the itegral will be ) Therefore therefore ψ Kψ3 3 K ψ 5 5 K3 ψ 7 37 R = (ρ Kρ3 K ρ K3 ρ 7 ) 3 7 ; R πab will express the probability that the values of a ad b are cotaied betwee these limits ( s A ± ρ ) ( ad s B ± ρ ) that is to say that the facilities of the errors which are beig foud repeated α ad β times which are proportioal to a s ad b s are ot separated from the quatities A ad B give by the observatios of a quatity greater tha ρ 5

26 If oe makes for greater simplicity ρ = µ AB oe will have K = µ ρ because A B = ad the probability i questio will be expressed i this maer ( µ µ3 π 3 µ5 4 5 µ 7 ) 46 7 Therefore if oe supposes µ = oe will have the series of which the sum is very early ; so that the sought probability will be early 648 π = Thus oe would i this case wager with advatage that i supposig the facilities of errors respectively equal to A ad B oe will ot be deceived by the quatity AB µ which because very large is ecessarily ifiitely small It would be much more difficult to fid the value of R if the variables ξ ψ ζ were more tha two especially because the itegratio must be such that it embraces oly the values of these same variables which are cotaied betwee the limits ρ ad ρ; but oe could i this case serve oeself by the approximatio that we have give i the precedig umber For this oe will remark that sice we have made r = ρ ad that is supposed very great the umber r must be very great also; so that oe will have very early T = r m 3 (m ) [ m m m(m ) m m(m ) (m 4) m by cotiuig this series util some oe of the umbers m m 4 becomes egative: therefore oe will have ( ) m ρ m m m(m ) m m(m ) P T = π (m 4) m 3 (m ) ABC ad there will be besides oly to multiply this quatity by W that is to say by the mea value or if oe wishes by the smallest value of V Now sice oe has V = e ( ξ A ψ B ζ C ) it is clear that the smallest value of V will be that where the quatity ξ A ψ B ζ C will be the greatest; ad it is easy to see that this will happe by takig ξ = ρ ψ = ρ ζ = 0 because ξ ψ ζ = 0 ad supposig that A ad B are the least of all the quatities A B C ; thus oe will have W = e ( A B )ρ 6 ]

27 Therefore makig for brevity M = mm m(m ) m m(m ) (m 4) m 3 (m ) oe will have the quatity M ( ρ π ) m e ( A B )ρ ABC which will be ecessarily less tha the sought probability; so that by amig H this quatity oe would always wager with advatage H agaist H that i supposig the facilities of the errors equal respectively to A B C oe will ot be deceived by the very small quatity ρ LEMMA I 3 Let X be a fuctio ratioal ad without divisor of x : oe demads the coefficiet of the power x µ i the series resultig from the developmet of the fractio X (a x) Oe has as oe kows (a x) = a x ( )x a a = 3( ) a 3x a 34()x a ; 3 ( ) therefore if oe orders the quatity X with respect to the powers of x by begiig with the highest i a maer that oe has i geeral X = Ax α Bx α Cx α Mx µ Nx µ ad if oe multiplies this series by that which expresses the value of (a x) it is easy to see that the term which will cotai the power x µ will be 3( ) a M 3 a N 34() 3 ( ) a P x µ so that the sought coefficiet will be represeted by the series 3( ) a M 3 a N 34() 3 ( ) a P We deote by X the sum of all the terms of the value of X where the powers of x are ot higher tha x µ so that oe has X = Mx µ Nx µ P x µ ; 7

28 dividig by x µ oe will have X x µ = M x N x P x 3 therefore differetiatig times ad makig ext x = a oe will have ( ) d X x ± µ 3 ( ) dx = a M 3 N a the upper sig beig for the case where is odd ad the lower sig for the oe where is eve Therefore this sought coefficiet of the power x µ will be equal to that which becomes the quatity ( ) 3 ( ) d ( dx ) X x µ whe oe makes x = a 4 REMARK If oe divides the quatity X by x µ ad if oe by rejectig ext all the terms where there will be positive powers of x it is clear that oe will have the value of X x µ ; therefore i the place of the quatity X oe ca have the same quatity X by havig care to reject the terms of which we happe to speak; i this maer oe will have for the expressio of the sought coefficiet of x µ the quatity ( ) d ( ) X x µ 3 ( ) dx by rejectig i this quatity before or after the differetiatios all the positive powers of x ad makig ext x = a 5 COROLLARY Suppose that oe demads the coefficiet of x µ i the series x α x x x 0 x x x β raised to the power Followig the ordiary rules of the summatio of the geometric progressios oe fids that the sum of this series is represeted by x α ( x αβ ) x so that the th power of the same series will be equal to x α ( x αβ ) ( x) Comparig therefore this formula with that of the precedig Lemma oe will fid X =x α ( x αβ ) =x α x ( )αβ ( ) x ( )α(β) ( )( ) x ( 3)α3(β) ; 3 8

29 therefore dividig by x µ ad makig for brevity oe will have α µ = π α β = ρ X x µ =x (π) x (π ρ) ( ) x (π ρ) ( )( ) x (π 3ρ) 3 ad cosequetly by differetiatig times ( ) d ( X x µ ) dx =(π )(π ) (π )x (π) (π ρ)(π ρ) (π ρ)x (π ρ) ( ) (π ρ)(π ρ) (π ρ)x (π ρ) Oe will reject therefore from this series the terms where the expoets of x will be foud positive that is to say if s is the whole umber which is equal or immediately greater tha π ρ oe will cotiue the series oly util the s th term; or else it will suffice to cotiue util some oe of the factors π π ρ become egative; ext oe will make x = ad oe will divide the whole by 3 ( ); oe will have thus the value of the sought coefficiet which will be cosequetly [(π )(π ) (π ) 3 ( ) (π ρ)(π ρ) (π ρ) ( ) (π ρ)(π ρ) (π ρ) ( )( ) (π 3ρ)(π 3ρ) (π 3ρ) 3 ] Thece oe obtais the solutio of the followig Problem PROBLEM VII 6 Oe has several observatios i each of which oe supposes that oe has bee able to be deceived equally by ay oe of these quatities α 0 β; oe demads what is the probability that the error of the mea result of observatios will be µ p or that it will be cotaied betwee these limits ad q 9

30 I order to fid the probability that the mea result be µ it is ecessary to seek the coefficiet of the power x µ of the polyomial x α x x x 0 x x x β raised to the power ad to divide ext this coefficiet by the value of the same polyomial raised to the power which correspods to x = that is to say by (α β ) ; this is that which follows evidetly from that which we have demostrated i the precedig Problems Therefore by the precedig Corollary oe will fid that the sought probability will be by makig π = α µ ρ = α β [(π )(π ) (π ) 3 ( )ρ (π ρ)(π ρ) (π ρ) ( ) (π ρ)(π ρ) (π ρ) ] by cotiuig this series util some oe of the factors π π ρ becomes egative Such is the geeral expressio of the probability that the mea error of observatios be µ ; thus i order to have the probability that the error be cotaied betwee the limits p q ad there will be oly to vary µ i the precedig quatity ad to take the sum of all the particular quatities which correspod to µ = p 0 q Now sice the quatity µ eters oly ito the value of π there will be therefore oly this variable quatity; so that the difficulty will be reduced to sum some sequece of which the geeral term will be of this form (s )(s )(s 3) (s k) For this let the sum of this series be represeted by u(s )(s )(s 3) (s k) u beig a ukow fuctio of s ad puttig s i place of s ad u i the place of u oe will have u s(s )(s ) (s k ); this quatity beig subtracted from the precedig oe will have the differece [u(s k) u s](s )(s ) (s k ); but it is ecessary that this differece be equal to the geeral term of the series of which oe seeks the sum therefore oe will have the equatio u(s h) u s = s k 30

31 to which oe will satisfy by makig u = s k k ; so that the geeral sum of the series of which the geeral term is (s)(s) (sk) will be represeted by (s )(s ) (s k)(s k ) k ad cosequetly the sum of all the terms cotaied betwee these two here will be equal to (s )(s ) (s k) ad (s )(s ) (s k) s (s )(s ) (s k) (s )(s ) (s k ) k Applyig therefore this to the formula foud above oe will have for the probability that the mea error falls betwee p ad q the followig expressio i which I have made for brevity α p = δ ad α q = γ [γ(γ ) (γ ) (δ )(δ ) (δ ) 3 ρ [(γ ρ) (γ ρ ) (δ ρ ) (δ ρ )] ( ) [(γ ρ) (γ ρ ) (δ ρ ) (δ ρ )] ] This series must be cotiued util some oe of the factors γ ρ γ ρ becomes egative; ad as for the other factors δ ρ δ ρ if ay oe of amog those are foud egative the it will be ecessary to augmet the umber δ by as may uits as it will be ecessary to reder it positive; this follows evidetly from this that the series of which the precedig is the sum must be cotiued oly util some oe of the first factors π ρ π ρ becomes egative as we have see i the precedig umber 7 COROLLARY Suppose that the umbers α ad β become ifiite as well as p ad q but i a way that they have betwee them some fiite ratio; ad let so that oe has β α = l p α = r q α = s β = αl p = αr q = αs l r s beig some fiite umbers; i this case oe will have ρ = α β = ( l)α δ = α p = ( r)α γ = α q = ( s)α; 3

32 so that by substitutig these values ito the precedig formula ad eglectig that which oe must eglect because α = oe will have this here where f = l [ ( s) ( s f) 3 f ( r) ( r f) ( ) ( s f) ( ) ( r f) each of these two series before are cotiued oly util ay oe of the quatities s f s f ad r f r f become egative The case of this Corollary holds whe oe supposes that each observatio is equally subject to all the possible errors cotaied betwee the give limits; because if oe takes the greatest egative error for uity ad if oe desigates the greatest positive error by l the precedig formula will deote the probability that the error of the mea result of observatios is cotaied betwee these two limits r ad s I the rest we will give further below a method much more simple to solve these sorts of questios PROBLEM VIII 8 Supposig that the errors which oe ca commit i each observatio are ω 0 ω ad that the umber of cases which correspod to each of these errors is proportioal respectively to 3 α 3 oe demads what is the probability that the error of the mea result of m observatios be cotaied betwee the limits p m ad q m We begi by seekig the probability that the mea error be µ m ; this probability will be equal to the coefficiet of the power x µ of the polyomial x ω x ω ωx (ω )x 0 ωx x ω x ω raised to the power m this coefficiet beig ext divided by the value of the same polyomial raised to the power which correspods to x = Now oe has ( ) x x (ω )x ω x ω x ω = (x x ω ) ω = ; x therefore the polyomial i questio will be equal to ( ) x x ω ω x ad cosequetly the power of this polyomial will be represeted by x mω ( x ω m ) ( x) m This formula beig compared to that of No 5 oe will have = m α = mω α β = ω ] 3

33 whece oe obtais = m α = mω = ω ad β = ω ; therefore (precedig Problem) the sought probability will be [(π )(π ) (π m ) 3 mρm m(π ρ)(π ρ) (π m ρ) m(m ) (π ρ)(π ρ) (π m ρ) ] by supposig π = mω µ ad ρ = ω ad cotiuig the series util some oe of the factors π ρ π ρ becomes egative Thece oe will fid as i the precedig Problem that the probability that the mea error is foud betwee the limits p ad q will be expressed by [γ(γ ) (γ m ) (δ )(δ ) (δ m) 3 mρm m[(γ ρ) (γ m ρ) (δ ρ) (δ m ρ)] m(m ) [(γ ρ) (γ m ρ) (δ ρ) (δ m ρ)] ] γ beig = mω q ad δ = mω p I regard to the cotiuatio of these two series it will be ecessary to follow the rules prescribed above (4) 9 COROLLARY Suppose ow that the umbers ω p ad q become ifiite but so that oe has p ω = r q ω = s r ad s beig some fiite umbers ad the precedig formula will become (5) (m s) m m[(m ) s] m m(m ) [(m ) s] m 3 m m (m r)m m[(m ) r] m m(m ) [(m ) r] m 3 m m these two series beig cotiued util some oe of the quatities which are raised to the power become egative This formula will express therefore the probability that the mea error of observatios is cotaied betwee the limits r ad s i the hypothesis that each observatio is subject to all the possible errors cotaied betwee these two limits ad ad that the facility of each error is proportioal to the differece that there is betwee this error ad the greatest possible error i the same sese; this hypothesis is more coformed to ature tha that of No 7; the curve of errors (0) would be here ay isosceles triagle 30 SCHOLIUM I geeral oe could fid by aid of the precedig Lemma the probability that the mea error is equal to a give quatity uder the hypothesis that the 33