280 THIRD REPORT— 1833. 



The coefficients Xp Xj, Xg, &c., are reciprocal* functions, pos- 

 sessing the following remarkable property, that / X^ X„ rf a: 



= 0, in all cases, unless n = m, in which case / X„X„ d x 

 1 ^-' 



2n + l' 



The knowledge of this property will readily enable us to de- 

 termine the following four different values of s : 



, , ab a'^b^ a-'b^ „ 



Sj = 1 + -g- H g 1 « 1- inc. 



_ \ a a^ a^ o 



1 6 6^ 6^ 



a ■'' 3 a^ + 5 «3 + 7 „ 



~ - ± + -i- + — ^- + — ^- + &c 

 "^^ ~ a 6 "^ 3 o^ 62 ^ 5 a^ 63 ^ 7 a^ 6^ + ^c. 



Whatever be the relation of a and b to each other and to 1, 

 two of these four series are convergent, and two of them di- 

 vergent. But it appears from the examination of the finite in- 



tegral / K K' t/ x, that one only of these two convergent 



series gives the correct value of z, being that which arises from 

 the combination of the two convergent developements of K and 

 K'', whilst the incorrect value arises from the combination of a 

 convergent developement of K with a divergent developement 

 of K', or conversely. The conclusion which is drawn from this 

 fact is, that the introduction of the divergent developement of 

 K or of K' vitiates the corresponding value of z, even though 

 that value is expressed by a convergent series. Let us now 



examine how far the definite integral of / K K' </ a: will jus- 

 tify such an inference. 



If Ave denote K K' by — , we shall easily find, 



• Functions which possess this property have been denominated reciprocal 

 functions by Mr. Murplijf, in a second memoir on the Inverse Method of Defi- 

 nite Integrals, in the fifth vohimeof the Transactions of the Philosophical Society 

 of Cambridge, in which general methods are given for discovering all species of 

 such functions, and where one very remarkable form of them is assigned. The 

 functions referred to in the text were first noticed by Legendre, in his fii-st me- 

 moir on the Attraction of Ellipsoids, and subsequently, at great length, in the 

 Fifth Part of his Exercices du Calciil Integral. Cauchy has used the term recipro- 

 ea/ function in a different sense ; see Exercices des Mathematiqtces, torn. ii. p. 141. 



