350 KEPORT— 1872. 



a 



ai(m,+ i,....)=Ce~'^'^''^"''+^'')Ai(«,. ...)«?(«,. ...)„; 



whence 



a 



Now put «j= -Kj ; then, since (Crelle, xlvii. p. 301), 



M(-k^, -K,....)=Al(K,,i,....), wehaveCi""^"=l, C,= ^. 

 and therefore 



Al(t/, + K,....) = e V -/Al(M^....)aZK....)„; 



whence 



Al(«,-K,....) = e V ^/A1(m,....)«?K....)„- 



"We will next investigate the value u^ assumes when v^ becomes 



v^ — m^TT + Sji^, .... (see Crelle, xlvii. p. 305). 

 It is plain that tt^ becomes 



and 2,Ki, A = 2.Ki, , ^^S^^ ^ = 2,K,, , Vm^ A. .K',, ^ . .r 



and therefore the required value of u^ is (mjm^ .... r r . . . . being here 

 or 1) 



^ -K^i i + m^Ki 2+ . . • ■) + Kr^'^'i,i + r,'K'i,2+ ■ ■ • .)=u,-K^ (Weierstrass, I c). 

 Section 12. — Hence 



g.e' ' '• ^ ^ ''aik-k,....). 



ISTow E(Mj-E:„ m,-K ) 



= IS AS,G,„ ^J,, ^(H, - i J(„^ - K J 



1 



