340 REPORT — 1872. 



In this expression the value (c) is excluded from those we successively give 



to 1'. 



To prove this last formula T refer the reader to a formula proved by 

 Weierstrass at the end of the first chapter of his memoir in the 52nd volume 

 of Crelle's Journal. Making use of a formula which will be found at p. 312 

 of the same volume, he gives 



1 P.r. dx' I P'a„ a — a 



[P'rt„ a-a^ 



Now substitute in the first member of this equation (.r^ — a„) — (a — «„) for 

 x^ — a, and in the second member (« — «a) — (^^""s) ^°^ '^'■~'^a' expand both 

 members in terms of a — cig, and equate the coefiicients of the first power of 

 a — n„ on both sides of the expression thus developed, and we have an equa- 

 tion of the form 



where S' applies to a, the value j3 being excluded from the values of a thus 

 arising, and q^, q„ certain constants depending respectively on a^ and t(„. 

 Allowing for the diiferent notation, this formula is equivalent to the equation 

 we wish to prove. 



Section 4. — Diff'erentiating equation (3) of last section, we have 



d . aXjn^u.^ ■ • ■ ■)c _ ___ ''\-i^\-i «?'K- • • •)2.-i . 



du^ «2o- 1 - «2^- 1 " «^'(Wl )2c-l 



From this we deduce, by applying formulae (1) and (2) of the first section, 



d '^"-i d 2a- 1 

 ^aX(^-K,....X = ;^aXK-K,....)„ (A) 



du '*'^v-i ^^■••••^.-rfu.. 



"We next put 



and also 



2c— 1 2c _ 2c-2 2c- 1 



J =J -J , ?J =J -J, 



J'.,c=J..i+J..2+----+J.,c; 



then the following equation is given connecting the new transcendents ; 



cHog al(u.,U„....) a a 2v-\ 2v-\ 



^^ ^ ^ ^ — ^'^ = J^-«\(M, + K,-K,....X + a\K-K,....V (1) 



duv 



If we write it 



dloa; al(uM^ ), , 2a- 1 2a-l 2,/-l 2v-l 



^' ^^; ^^ = J.-«\(«, + K.-K,....).+«K«.-K,....X 



