1 62 Dr. James Bottomley ^;^ 



(25) we shall get 



1 = (Ui + 2V2X + 3U3^2 ^ 4xj^^ + ^ <fec.) 



(Ao + Ai^ + A2^2 + A3a;^ + &c.); (30) 



effecting the multiplication, and arranging the result in 

 ascending powers of x we shall obtain 



l=AoUi + a7(Ao2U2 + AiUi) 



+ ^^(AoSEe + A12U2 + A2U1) + x%Ao4:V, + AiBUs + A22U2 + A3U1) 



+ a^*(Ao5U5 + Ai4U4 + AaSUs + A32U2 + A^Ui) 



+ ^(AoGUs + Ai5U5 + A24U4 + A33U3 + A,2U2 + A5U1) 



+ , &c. ; (31) 



as this equation holds for all values of ;ir we shall have 



AoUi=l (32) 



2U2Ao + AiUi = (33) 



3U3Ao + 2U2Ai + UiA2 = (34) 



4U4A0 + 3U3A1 + 2U2A2 + U1A3 = (35) 



SUgAo + 4U4A1 + 3U3A2 + 2U2A3 + U1A4 = (36) 



The general equation being 



.nVn^o + (n - l)U„_iAi + (n - 2)U.«_2A2 + &c. + UiA„_i = (37) 

 From (32) we obtain 



A _ 1 



substituting this value in (33) we obtain 



substitute these values of A^, Aj, in (34), then 



4U2^ 3U3 



these values of Aq, Ai, A2, being substituted in (35), then 



, _ 4U4 I2U3U2 8U2^ 



W Ui^ W " 

 substitute these values of Aq, Aj, A2, A3 in (36), then 

 . ^_5U5 I6U4U, 36U3U2^ 9U32 IGUa^ 



By a continuation of this process the values of the 



