







j (m, X) = 



- 





that of 





















( 1 \'' 



- + X 



r 1 



1 2/ 



\ n-r 



or, 



what 



is 



the 



same thing, of 



(jj-^T + X iy-^y 



i-r 



-\ = 



and on Co-resolvents, 189 



either with Boole's results or with obvious properties of the 

 ?^-ary resolvent of the equation in y or y''^. If the general 

 n-SiYj resolvent of (v) be denoted by 



(xxii) 

 = - (xxiii) 



(xxiv) 



will still be (xxii), for (xxiii) is but the result of the divi- 

 sion of (v) by 2/". Hence if in (xxiv) we replace y'^ by z 

 and xhy — x, we must replace "tnhj — "ni and cc by — x in 

 (xxii) in order to obtain the '?i-ary resolvent of 



z'^ — xz''-^ — 1=0 - - - (xxv) 



that resolvent is, consequently, 



/ ( — m, — x) = - - - (xxvi) 



m other words when, in (v) we change r into n — r the 

 index of x in the resolvent remains unaltered. 



27. The demonstration of the generalized theorem is as 

 follows. Let 



y'^ — xy^ — 1 = - - (xxvii) 



then by Lagrange's theorem, employed precisely as Boole 

 has done (ibid. p. 735), we find that u or 2/"* may be ex- 

 panded in a series of the form 



Uq + UiX + u^x^ + &c. ad. inf., 



in wliich 



\ m + ar ^ f'^ /-.x '-^ 

 m — 1 X (1) " 



u^= r— - (xxvm) 



n[TY ^ ^ 



and consequently 



\ m + ar ^ Y'^ /-.x^^ 

 ^^i I — ^^ - 1 j • (1) ^ 



[r]«U^= r- - (xxix) 



