Leonhard Euler THEOREM 1. x f 1 dx(1 x g ) n n DEMONSTRATION

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1 Epasio of the itegral forula f dlog havig eteded the itegratio fro the value = 0 to = Leohard Euler THEOREM If deotes ay positive iteger ad the itegratio of the forula f d g is eteded fro the value = 0 to = its value will be = g f f + g f + g f + g f + g. DEMONSTRATION It is kow that i geeral the itegratio of the forula f d g ca be reduced to the itegratio of this oe f d g sice it is possible to defie costat quatities A ad B i such a way that it is Origial title: Evolutio forulae itegralis f dlog itegratioe a valore = 0 ad = etesa first published i Novi Coetarii acadeiae scietiaru Petropolitaae pp. 9-9 reprited i Opera Oia: Series Volue 7 pp. 6-7 Eeströ-Nuber E traslated by: Aleader Aycock for the project Euler-Kreis Maiz

2 f d g = A f d g + B f g ; for havig take differetials this equatio arises f d g = A f d g + B f f d g Bg f +g d g which divided by f d g gives g = A + B f g Bg g or g = A Bg + B f + g g ; that this equatio ca hold it is ecessary that it is whece we coclude = B f + g ad A = Bg B = f + g ad A = g f + g. Therefore we will have the followig geeral reductio f d g = g f + g f d g + f + g f g ; because it vaishes for = 0 if f > 0 of course the additio of a costat is ot ecessary. Hece havig eteded both itegrals to = the last itegral part vaishes by itself ad for the case = it will be Sice for = it is f d g = g f + g f d g 0 = f f = f f d g. havig put = we obtai the followig values for the sae case =

3 f d g = g f f d g = g f f d g = g f f + g f + g f + g f + g f + g f + g ad hece for ay positive iteger we coclude that it will be f d g = g f f + g if oly the uber f ad g are positive. f + g f + g f + g COROLLARY Hece vice versa the value of a product of this kid fored fro a arbitrary aout of factors ca be epressed by a itegral forula so that it is f + g f + g f + g f + g = f g havig eteded this itegral fro the value = 0 to =. f d g COROLLARY So if oe therefore has a progressio of this kid f + g f + g f + g f + g f + g f + g f + g f + g f + g f + g etc. its geeral ter that correspods to the idefiite ide is coveietly represeted by this itegral forula f g f d g by whose eas the progressio ad its ters correspodig to fractioal idices ca be ehibited.

4 COROLLARY If istead of we write we will have f + g f + g f + g f + g = f g f d g which ultiplied by f +g yields f + g f + g f + g f + g = f g g f + g f d g. REMARK It would have bee possible to derive this last forula iediately fro the precedig oe sice we just proved that f d g = g f + g f d g if both itegrals are eteded fro the value = 0 to = ; this is to be kept i id about the deteriatio of the itegrals i everythig that follows. Furtherore it is also always to be oted that the quatities f ad g are positive which coditio the give proof deads of course. Cocerig the uber if by it oe deotes the ide of a certai ter of the progressio there is othig to ipede that either ay positive or egative ubers are deoted by it because all ters also correspodig to egative idices of its progressio are cosidered to be ehibited by the give itegral forula. Nevertheless it is carefully to be oted that this reductio f d g = g f + g f d g is oly true if > 0 because otherwise the algebraic part f +g f g would o vaish for =. REMARK 6 Series of this kid which ca be called trascedetal because the ters correspodig to fractioal idices are trascedetal quatities I already oce studied i Coet. acad. sc. Petrop. book i ore detail;

5 hece i this place I did ot scrutiise those progressios as the rearkable coparisos of the itegral forula which are derived fro there ore diligetly. After I had show that the value of the idefiite product is epressed by the itegral forula d log eteded fro = 0 to = which if is a postive iteger by itegratio itself is aifest I eaied the cases i which a fractioal uber is take for ; there it is ideed aythig but clear fro the itegral forula itself to what kid of trascedetal quatities these ters are to be referred. But by a sigular artifice I reduced the sae ters to better-kow quadratures what therefore sees ost worthy to be cosidered i ore detail. 7 Sice it was deostrated that PROBLEM f + g f + g f + g f + g = f g f d g havig eteded the itegral fro = 0 to = to assig the value of the sae product i the case g = 0 by eas of a itegral forula. SOLUTION Havig put g = 0 i the itegral forula the eber g vaishes but at the sae tie also the deoiator g vaishes whece the questio reduces to that that the value of the fractio g g is defied i the case g = 0 i which so the uerator as the deoiator vaishes. For this ai let us cosider g as a ifiitely sall quatity ad because g = e g log it will be g = + g log ad hece g = g log = g log ; fro there for this case our itegral forula chages ito f f d log so that oe ow has or f = f = f + f d log f + d log.

6 COROLLARY 8 As ofte as is a positive iteger the itegratio of the forula f d log succeds ad havig eteded it fro = 0 to = ideed the product arises that we foud to be equal to this forula. But if fractioal ubers are take for the sae forula will serve for iterpolatig this hypergeoetric progressio or etc etc. COROLLARY 9 If the epressio just foud is divided by the pricipal oe a product will arise whose factors proceed i a arithetic progressio f + g f + g f + g f + g = f g f d log f d g whose values ca also if is a fractioal uber ca be assiged fro there. 0 Because it is f d g = COROLLARY g f + g it will also i the sae way be for the case g = 0 f d log = f ad hece by those other itegral forulas ad = f f d g f d log f d log 6

7 f + g f + g f + g = f g f d log f + g f d g. REMARK Because we foud that = f + f d log it is clear that this itegral forula does ot deped o the value of the quatity f what is also easily see by puttig f = y whece it becoes f f d = dy ad log = log = f log y = f log y ad therefore f log = log y so that it is = dy log y which forula arises fro the first by puttig f =. For a iterpolatio of this kid the whole task is therefore reduced to that that the values of the itegral forula d log are defied wheever the epoet is a fractioal uber. As for eaple whe = oe has to assig the value of the forual d log which value I already oce showed to be = π while π deotes the circuferece of the circle whose diaeter is = ; but for other fractioal ubers I taught to reduce its value to quadratures of algebraic curves of higher order. Because this reductio is by o eas obvious ad is oly valid whe the itegratio of the forula d log is eteded fro the value = 0 to = it sees to be worth of special attetio. But eve though I already treated this subject oce I evertheless because I was led there o a quite o straightforward way decided to resue the sae here a epad it i ore detail. 7

8 THEOREM If the itegral forulas are eteded fro the value = 0 to = ad deotes a positive iteger it will be = g f +g d g f d g f d g o atter which positive ubers are take for f ad g. Because above we showed that DEMONSTRATION f + g f + g f + g = f g g f + g f d g we will if we write istead of have f + g f + g f + g = f g g f + g f d g. Now divide the first equatio by the secod ad this third oe will arise f + + g f + + g f + g + + = g f + g f f + g d g f d g. But if i the first equatio istead of f oe writes f + g this fourth equatio will arise f + + g f + + g f + g = f + gg g f + g f +g d g. Multiply this fourth equatio by that third oe ad oe will fid the equatio to be deostrated itself = g f +g d g f d g f d g. 8

9 COROLLARY If i the first equatio oe sets f = ad g = the sae product will arise + + = d g havig copared which eqautio to that oe we obtai d g f f +g d g = d g f d g. COROLLARY If we write g istead of i that equatio it will be + + = g g d g so that we reach this copariso betwee the followig itegral forulas g d g = f +g d g f d g f d g. COROLLARY If i the equatio of the theore we put g = 0 because of g = g log the powers of g will cacel each other ad this equatio will arise + + = f d log f d log f d log whece we coclude f d log f d log = g g d g or because of 9

10 f d log = f f d log this oe f f d log f d log = g g d g. COROLLAR 6 Let us put f = g = ad = here that is a positive iteger ad because of it will be ad hece d log = d log = d log = ad by takig = because of d = π d d oe will have d log = d = π. 0

11 REMARK 7 So behold this succict proof of the theore oce propouded by e that d log = π ad behold that I did ot use a arguet ivolvig iterpolatio which I had used the. Here it was of course deduced fro this theore i which I foud that f d log f d log = g g d g. But the pricipal theore whece this oe is deduced behaves as follows g f d g f +g d g = f d g d ; for each eber by itegratio eteded fro = 0 to = is epaded i this uerical product + +. Ad if we wat to assig a class etedig further to the other eber the theore ca be propouded so that it is g f d g f +g d g f d g = k k d k ad if here oe takes g = 0 it is f d log f d log = k k d k. Therefore it is especially to be oted that that equality holds o atther what ubers are take for f ad g; i the case f = g it is ideed clear because for it will be g d g = g g = g ;

12 ad because g g+g d g = k k d k g+g d g = g d g the equality is perspicuous because k ca be take ad libitu. But i the sae way o which we got to this theore it is possible to get to other siilar oes. THEOREM 8 If the followig itegral forulas are eteded fro the value = 0 to = ad deotes ay positive iteger it will be + + = g f +g d g f d g f d g o atter which positive ubers are take for f ad g. DEMONSTRATION I the precedig theore we already saw that f + g f + g f + g = f g g f + g f d g ; if i the sae way we write istead of i the pricipal forula we will have f + g f + g f + g = f g g f + g f d g fro where that equatio divided by this oe produces f + + g f + + g f + g + + = g f + g f f + g d g f d g.

13 But if we write f + g istead of f i the pricipal equatio we obtai this equatio f + + g f + + g f + g = f + gg g f + g f +g d g. Now ultiply this equatio by the precedig ad the equatio itself which is to be proved will arise + + = g f +g d g f d g f d g. COROLLARY 9 We obtai the sae value fro the pricipal equatio by puttig f = ad g = so that + + = d which itegral forula by writig k istead of is trasfored ito this oe so that k k d k g f +g d g f d log f d log = k k d k. COROLLARY 0 If we set g = 0 here because of g = g log equatio we will have this f d log f d log f d log = k k d k ;

14 because we had foud before f d log f d log = k k d k by ultiplyig the by each other we will have these equatios f d log f d log = k k d k k d k. COROLLARY Without ay restrictio oe ca put f = here; the for = ad k = it will therefore be d log d log 0 = 9 ad because of d d d log = d log ad d log 0 = d log = d d ; but the for = ad k = it will be or d log d log = 9 d d d log = d d.

15 GENERAL THEOREM If the followig itegral forulas are eteded fro the value = 0 to = ad deotes ay positive iteger it will be λ + λ + λ + = λ λ + g f +λg d g o atter which positive ubers are take for the letters f ad g. f d g λ f d g λ+ Because as we showed above it is DEMONSTRATION f + g f + g f + g = f g g f + g f d g if we write λ istead of here at first but the λ + we will obtai these two equatios λ f + g f + g f + λg = f λg g λ f + λg λ + f + g f + g f + λ + g = f λ + g g λ+ f + λ + g of which that oe divided by this oe yields f + λg + g f + λg + g f + λg + g λ + λ + λ + f d g λ f d g λ+ λ f + λg + g f = g λ + f + λg d g λ f d g λ+. But if we i the first equatio write f + λg istead of f we will obtai f + λg + g f + λg + g f + λg + g = f + λgg g f + λg + g f +λg d g which two equatios ultiplied by each other produce the equatio to be deostrated itself λ + λ + λ + = λg f +λg d g f d g λ λ + f d g λ+.

16 COROLLARY If we put f = λ ad g = i the pricipal equatio we will also fid λ + λ + λ + = λ λ + λ d which for by writig k istead of chages ito this oe λk λ + λk d k so that we have this very far-etedig theore g f +λg d g f d g λ f d g λ+ = k λk d k. COROLLARY This theore ow holds eve if is ot a iteger; let us eve because the uber λ ca be take ad libitu write istead of λ ad we will reach this theore f d g f d g + = k k d k g f +g d g. COROLLARY If we put g = 0 because of g = g log for this theore will take this f d log f d log + = k k d k f d log which is ore coveietly represeted as follows f d log f d log f d log + = k k d k where its evidet that the ubers ad ca be perutated. 6

17 REMARK 6 So we foud two sources whece it is possible to scoop iuerable coparisios of itegral forulas; the oe source opeed up i cotais itegral forulas of this kid p d g q which I already treated soe tie ago i the observatios o the itegral forulas p d q eteded fro the value = 0 to = where I showed at first that the letters p ad q ca be couted that it is p d q = q d p but the that p d p = π si pπ ; But especially I deostrated that p d q p+q d = r p d r p+r d q i which equatio the copariso foud i is already cotaied so that fro this othig ew I did ot already epad ca be deduced. Therefore I here aily attept to ivestigate the other source idicated i ; sice without ay restrictio oe ca take f = our priary questio will be d log d log d log + = k k d k by eas of which the values of the itegral forula d log λ wheever λ is ot a iteger ca be reduced to quadratures of algebraic curves; sice as ofte as λ is a iteger oe ca carry out the itegratio because 7

18 d log λ = λ. But the questio of greatest iportace is about the cases i which λ is a fractioal uber. Therefore I will defie these for the kids of the deoiatio successively here. PROBLEM 7 While i deotes a positive iteger to defie the value of the itegral forula d log i havig eteded the itegratio fro = 0 to =. SOLUTION I our geeral equatio let us put = ad it will be d log d log = k k d k. Now let be = i ad because of = i + it will be d log = i + ; ow further take k = that k = i + ad it will be ad hece log d i i + = log d i = i + i+ d i i+ d i where it is evidet that it is coveiet to take oly odd ubers for i because for the eve oes the epasio is aifest per se. 8

19 COROLLARY 8 But all cases are easily reduced to i = or eve to i = ; for as log as i + is ot a egative uber the reductio foud holds. For this case it will therefore be because of d = π. d log = d = π COROLLARY 9 But havig settled this pricipal cases because of we will have ad i geeral d log = d log = π d log + d log d log = π = 7 + π. PROBLEM 0 While i deotes a positive iteger to defie the value of the itegral forula d log i havig eteded the itegratio fro = 0 to =. SOLUTION Let us start fro the equatio of the precedig proble d log d log = k k d k 9

20 ad let us put = i the geeral forula that oe has d log d log d log = k k d k ad by ultiplyig these two equatios we obtai d log d log = kk k d k k d k. Now just put = i here that d log i = i ad take k = ad it will arise log d i i = 9 i d i i d i whece we coclude log d i = 9 i i d i i d i COROLLARY Here two pricipal cases occur o which all reaiig deped of course by puttig either i = or i = which are I. II. d log = 9 d log = 9 d d d d 0

21 which last forula because of d = d chages ito d log d = d COROLLARY If we as i y observatios etioed before for the sake of brevity put ad as there for this class but the it will be p d p = q q = π si π = = α d = A I. II. d log = 9 d log = COROLLARY = 9αA αα = A. For the first case we will therefore have

22 d log = 9αA d log = 9αA ad d log but for the other case o the other had + = 7 + 9αA d log = αα A d log = αα A ad d log = 8 αα A. PROBLEM While i deotes a positive iteger to defie the value of the itegral forula d log i havig eteded the itegratio fro = 0 to =. SOLUTION I the solutio of the precedig proble we were led to this equatio d log d log = kk k d k but the geeral forula by takig = yields k d k ; d log d log d log = k by cobiig which forulas we obtai k d k

23 d log d log = k k d k Let = i ad take k = ad it will be k d k k d k. i d log = i i d i i d i i d. i COROLLARY So if i = we will have d log = d d d ; if this epressio is deoted by the letter P it will be i geeral d log = 9 P. COROLLARY 6 For the other pricipal case let us take i = ad it will be d log = d d 8 d or after a reductio to sipler fors d log = 8 d d d ; if the epressio is deoted by the letter Q it will be i geeral d log = 7 Q.

24 REMARK 7 If we idicate the itegral forula p d by the sig q solutio i geeral will behave as follows d log i i = i i ad for the two cases epaded before P = ad Q = 8 Now let us put for the forulas depedig o the circle = π si π = α ad but for trascedetal oes of higher order d = = = o which all reaiig deped ad we will fid whece it is clear that π si π i i d = A P = αα AA ad Q = ααβ β AA PQ = α = π si π. But because it is α = π ad β = π it will be i i. = β P = π πaa ad Q = 8AA ad P Q = A. π PROBLEM p q the 8 While i deotes a positive iteger to defie the value of the itegral forula log d i havig eteded the itegratio fro = 0 to =.

25 SOLUTION Fro the precedig solutios it is already perspicuous eough that for this case oe will fially reach this forula i d log = i i d i i d i i d i i d i which itegral forulas are to be referred to the fifth class of y dissertatio p etioed above. Hece if i they sae way as there the sig q deotes this forula p d the value looked for ca be ore coveietly q epressed so that d log i = i i i i i i i i i where it ideed suffices to have assiged values saller tha five to i; but wheever the uerators eceed five it is to be oted that but the further i + i i = + i = + i = + i i + + i i + i i + 0 i Furtherore for this class ideed two forulas ivolve the quadrature of the circle which are = π si π = α ad = π si π but two cotai higher quadratures which we wat to put. = β = d = d = A ad = d = B

26 ad fro these I assiged the values of all reaiig forulas of this class of course = = = = α = β A = β B = A = β = ββ αb ; = αb β = B; = αa β. = = ; = α A ; COROLLARY Havig take the epoet i = it will be d log = = α β A B whece we coclude i geeral that while deotes ay positive iteger d log = 6 α β A B. COROLLARY 0 Now let i = ad because it arises because of d log 6 = = = ad 6 8 = 8 6

27 it will be this epressio ad i geeral = αβ BB A d log = 7 αβ BB A. Let i = ad the for foud COROLLARY d log = 6 9 because of 6 = 9 = 7 = 7 0 chages ito = β α A BB whece it is cocluded i geeral d log = 8 β α A BB. Fially for i = our for COROLLARY d log = because of 7

28 8 = 7 = = 6 0 will be trasfored ito this oe so that it is i geeral 6 d log = ααββ AAB = 9 ααββ AAB. REMARK If we represet the value of the itegral forula d log λ by the sig [λ] the cases epaded up to ow yield [ ] = α β A B [ ] = αβ BB A [ ] = β β [ ] = α β A BB AAB [ + ] = α β A B [ + ] = αβ BB A [ + ] = β α A BB [ + ] = α β AAB whece by cobiig two whose idices added give 0 we coclude 8

29 [ + ] [ ] = α = π si π [ + ] [ ] = β = π si π [ + ] [ ] = β = π si π [ + ] [ ] π = α = si π. But fro the precedig proble we deduce i the sae way [ ] = P = αα [ ] = Q = ααβ β AA AA [ + ] = αα β AA [ + ] = ααβ AA ad hece [ + ] [ ] = α = π si π [ + ] [ ] = α = π si π whece i geeral we obtai this theore that it is [λ] [ λ] = λπ si λπ the reaso for which ca be give fro the iterpolatio ethod oce eposed as follows. Because it is it will be [λ] = λ λ + λ λ λ + λ λ λ + λ etc. 9

30 ad hece [ λ] = +λ λ +λ λ +λ λ etc. λ λ λ [λ] [ λ] = λλ as I deostrated elsewhere. λλ λπ etc. = 9 λλ si λπ PROBLEM 6 - GENERAL PROBLEM If the letters i ad deote positive itegers to defie the value of the itegral forula d log i or d log i havig eteded the itegratio fro = 0 to =. SOLUTION The used ethod will ehibit the value looked for epressed by eas of quadratures of algebraic curves i the followig way log d i = i i d i i d i i d i So if we for the sake of brevity deote the itegral forula p d by q p this character q but o the other had the forula log d by this [ ] [ so that ] deotes the value of this idefiite product z while z = the value looked for will epressed ore succictly as follows [ ] i i = i i whece it is also cocluded i i i i i i i 0

31 [ ] i = i i i i i i i i i i Here it will always be sufficiet to have take the uber i saller tha because it is kow for greater oes that i. [ ] i + = i + [ ] i i the sae way [ ] i + = i + i + [ ] i etc. ad so the whole ivestigatio is reduced to those cases oly i which the uerator i of the fractio i is saller tha the deoiator. I additio it will be helpful to have oted the followig o the itegral forulas p d p = : q q I. The letters p ad q are perutable that p = q q. p II. If oe of the two ubers p or q is equal to the epoet the value of the itegral forula will be algebraic of course p = = p p or q = = q q. III. If the su of the ubers p + q is equal to the epoet the value of p the itegral forula q ca be ehibited by eas of the circle because p = p p = p π si pπ ad q = q q = q π si qπ. IV. If oe of the ubers p or q is greater the the epoet the itegral p forula q ca be reduced to aother oe whose ters are saller tha what happes by eas of this reductio p + q = p p + q p. q

32 V. Aog ay of the itegral forulas of this kid there cosists such a relatio that p p + q p = q r r p + r q = q r q + r p by eas of which oe fids all reductios that I eposed i y observatios o these forulas. COROLLARY If i this way by eas of reductio IV we accoodate the foud forula to the sigle cases we will be able to ehibit the i the ost siple way i the followig aer. Ad for the case = i which o further reductio is eccessary we will have [ ] = = π si π = π. COROLLARY 6 For the cases = we will have these reductios [ ] = [ ] =. COROLLARY 7 For the case = oe obtais these three reductios [ ] = [ ] = =

33 because of = [ ] = because i the iddle it is = ] ] [ = [ ; = π it will be as before of course = π. COROLLARY 8 Now let be = ad these four reductios arise [ ] = [ ] = [ ] = [ ] =. COROLLARY 9 Let = 6 ad we will have these reductios [ ] = [ ] = 6 6 [ ] = = 6 = 6 6

34 [ ] = = 6 [ ] = COROLLARY 6 0 For = 7 the followig si equatios arise [ ] = [ ] = [ ] = [ ] = [ ] = [ ] 6 = COROLLARY 7 Now let be = 8 ad oe will get to these seve reductios [ ] = [ ] = = 8 6

35 [ ] = [ ] = = 8 [ ] = [ ] 6 = = [ ] 7 = REMARK It would be superfluous to epad these cases ay further because fro the oes listed the structure of these forulas is already see well eough. If i the propouded forula [ ] the ubers ad are prie to each other the law is aifest because [ ] = but if these ubers ad have a coo divisor it will ideed be useful to reduce this to the sallest for ad etract the searched value fro the precedig cases; evertheless the operatio ca also be eecuted as follows. Because the searched epressio certaily has this for [ ] = PQ where Q is the product of the itegral forulas P o the other had the product of soe absolute ubers for fidig that product Q just cotiue this series of forulas etc. util the uerator eceeds the epoet ad istead of this uerator write its ecess over ; if this is put = α that our forula is α this uerator α will give a factor of a product P; the fro there o further put the series of forulas α α+ α+ etc. util oe agai reaches a uerator greater tha the epoet ad the ;

36 +β β forula istead of which oe has to write ad fro there the factor β is iferred ito to product ad like this oe has to cotiue util forulas for Q will have arose. To uderstad these operatios i a easier way let us epad the case of the forula [ ] 9 = 9 PQ i this aer where the ivestigatio of the letters P ad Q is eecuted as follows: for Q for P ad so oe fids Q = 9 Because it is 9 = 9 it is PQ = 6 9 [ ] 9 = ad P = ad hece THEOREM No atter which positive ubers are idicated by the letters ad i the used otatio eplaied before it will always be [ ] =. DEMONSTRATION For the cases i which ad are ubers that are prie to each other the validity of this theore was evicted i the precedig theores; but that it also 6

37 holds if those ubers ad ejoy a coo divisor is ot evidet fro there; but fro this itself that for the cases i which ad are relatively prie the validity is cofired oe ca without doubt coclude that this theore is true i geeral. I certaily by o eas dey that this kid to coclude soethig is copletely sigular ad has to see suspect to ost people. Hece to leave o roo for doubt because for the cases i which the ubers ad are coposite we obtaied two epressios it will be useful to have show the aggreeet for the cases epaded before. But the cases = already provides a huge cofiratio i which case our forula aifestly produces the uity. COROLLARY The first cases deadig a deostratio of the agreeet is that oe i which = ad = for which we foud above 7 but ow via the theore it is where by copariso it is [ ] = ; [ ] = = whose validity was cofired i y observatios etioed above. COROLLARY If = ad = 6 it is fro the thigs above 9 [ ] = ; ow o the other had by eas of the theore 7

38 [ ] = ad therefore it has to be = whose validity is clear fro the sae source. COROLLARY 6 If = ad = 6 oe gets to this equatio = but if = ad = 6 it is i siilar aer or = = which is also detected to be true. ; COROLLARY 7 The case = ad = 8 yields this equality 6 = but the case = ad = 8 this oe = ad fially the case = 6 ad = 8 this equatio

39 6 6 6 which also agree with the truth. = REMARK 8 But if i geeral the ubers ad have the coo factor ad the propouded forula is [ ] [ = ] because it is [ ] = after havig reduced the sae to the epoet it will be 6 6. By the theore the sae epressio o the other had becoes whece for the epoet it will be 6 6 = If i the sae way the coo divisor is oe will fid for the epoet = which equatio ca be ore coveietly ehibited as follows. 9

40 = 6 7 But if i geeral the coo divisor is d ad the epoet d oe will have d d d d d d d d d d d d = d d d d d d d d which equatio ca easily be accoodated to ay cases whece the followig theore deserves it to be oted. d. THEOREM p 9 If α was a coo divisor of the ubers ad ad the forula q deotes the value of the itegral p d eteded fro = 0 to = it will q be α α α α α α α = α α. DEMONSTRATION Fro the precedig reark the validity of this theore is already see; while there the coo divisor was = d ad the two propouded ubers were d ad d here I just wrote ad istead of the but istead of their divisor d I wrote the letter α which kid of divisor the euciated equality cotais i such a way that oe assues the ubers ad ad therefore also α ad α to occur i the cotiued arithetic progressio α α α etc. I additio I a forced to cofess that this deostratio is of course aily based o iductio ad caot be see to be rigorous by ay eas; but because we are evertheless covicted of its truth this theore sees to be worthy of eve greater attetio; there is evertheless o doubt that a 0

41 further epasio of itegral forulas of this kid will fially give a perfect deostratio; but that it was possible for us to see its truth before hece shies a etraordiary specie of aalytical ivestigatio. COROLLARY 60 So if we istead of the used sigs substitute the itegral forulas itself our theore will behave as follows that α α α α α d α d α d = α d d d. COROLLARY 6 Or if we for abbreviatio set = X it will be α α α α = α α d X d X α d X d X α d X d X d X.

42 GENERAL THEOREM 6 If the divisors of the two ubers ad are α β γ etc. ad the forula p q deotes the value of the itegral p d eteded fro = 0 to = q the followig epressios fored fro itegral forulas of this kid will be equal to each other α α α α α α α α α β β β β β = β β β β γ γ γ γ γ = γ γ γ γ etc. DEMONSTRATION Fro the precedig theore the validity of this theore aifestly follows because every sigle of these epressios is equal to this oe which correspods to the uity as sallest coo divisor of the ubers ad. Therefore so ay epressios of this kid ca be equal to each other as there were coo divisors of the two ubers ad. COROLLARY 6 Because this forula is = ad hece = our equal epressios ca be ore succictly represeted as follows

43 α α α α α α β β β = β β β γ γ γ = γ γ γ α β γ. For eve if the uber of factors was icreased here evertheless the way of the copositio easily eets the eye. COROLLARY 6 So if = 6 ad = because of the coo divisors of these ubers 6 oe will have the followig for fors all equal to each other = = = = COROLLARY 6 If the last is cobied with the peultiate this equatio will arise = but the last copared to the pepeultiate yields =

44 REMARK 66 Hece ifiitely ay relatios betwee the itegral forulas of the for p d p = q q follow which are eve ore rearkable because we were led to the by a copletely sigular ethod. Ad if soeoe still doubts their truth he should cosult y observatios o these itegral forulas ad will the hece for ay case easily be coviced of their truth. But eve if that treatet serve for this cofiratio the relatios foud here are evertheless of eve greater iportace because i the a certai structure is oticed ad they are i easy fashio cotiued throughout all classes o atter how oe wats to assue the epoet whereas i the first treatet the calculatio for the higher classes becoes cotiuously ore labourious ad itricate. SUPPLEMENT CONTAINING THE DEMONSTRATION OF THE THEOREM PROPOUNDED IN It is coveiet to derive this deostratio fro the thigs etioed above; just take the equatio give i which for f = havig chaged the letters is d log ν d log µ d log ν+µ = κ ad by kow reductios represet it i this for κµ d κ ν d log ν d log µ d log ν+µ = κµν µ + ν Now set ν = ad µ = λ but the κ = that we have κµ d κ ν. d log d log λ = λ λ+ d log λ + λ d κ

45 which for the sake of brevity havig used the way fro above is ore coveietly epressed as follows [ ] [ λ ] [ λ+ ] = λ λ. λ + Now istead of λ successively write the ubers... ad ultiply all these equatios whose uber is = ad the resultig equatio will be = + [ ] [ ] [ ] [ ] [ ] [ + ] [ + ] [ + ] [ + ] = But i siilar aer just trasfor the first part that [ ] [ [ + ] [ ] [ ] [ ] ] [ + ] [ + ] [ + ]. whose agreeet with the precedig is revealed by cross ultiplicatio. But because it is fro the ature of these forulas [ ] + = + [ ] [ ] + = + [ ] [ ] + = + [ ] etc. because of these forulas this first part will becoe [ ] ; because this oe is equal to the other part ehibited before we obtai this equatio [ ] =

46 so that [ ] = which because = copletely agrees with the oe propouded i whece its truth was ow ideed evicted fro ost certai priciples. DEMONSTRATION OF THE THEOREM PROPOUNDED IN 9 Also this theore eeds a ore solid deostratio which I give fro the equatio established before [ ] [ λ ] [ λ+ ] = λ λ λ + as follows. While α is a coo divisor of the uber ad successively write the uber α α α etc. up to istead of λ whose aout is = α ad ow ultiply all equalities resultig i this aer that this equatio arises = α α + α [ ] [ α ] [ α ] [ α ] [ α ] [ +α ] [ +α ] [ +α ] [ + ] α + α α + α + α Now trasfor the first part i this oe equal to it α. [ ] [ α ] [ α ] [ α ] [ α ] [ +α ] [ +α ] [ +α ] [ + ] which because of [ ] [ +α = +α α ] ad so for the reaiig is reduced to this oe [ ] α + α + α + α + The secod part of the equatio o the other had is i siilar aer trasfored ito this oe 6

47 α α + α α + α whece this equatio arise ad hece [ α + α α + ] α α = α α α α α [ ] = α α α α α α α α α α α which equaio copared to the precedig yields this equatio α α α α α α α = what is to be uderstood about all coo divisors of the two ubers ad. 7