164 Dr. James Bottomley on 



,oo/^ (wyw Us^uA ..n/' wcw)^ UbU.^\ 



By (24), this expansion is equal to 

 log( - a) + cUo + x{c^x -h)-v cU2a:^ + cUao;^ + c\^^x^ + cUsa?^ + &c.; 

 hence if we equate the coefficients of corresponding powers 

 of X we shall obtain the following equations — 



logUi^ - logUi = log - a + cUo 



%U?"W/"^(W)"W>'~ ^ 



lf,«^U/ UA ,^/U3^W U3UA (mr UA\ ^, 



3|nw~uJ-Hiu?7""u7J^^lavp-w;r 



U .^W UA onr^^'W U4UA ,«/(IW W\ 



41 w'trj-^^iuiy - Ui^ 7~^\uiT-Ui^; 



i(on/^w U6\ ^^/UeW U5UA ../MM' u^w\ 

 -ou^ (Ui7 " UiV'^^^V (UiT W y 



" V~(wr" w J'^^^wr UiOj '^^' 



In a similar manner the relationship of the coefficients of 

 higher powers of x may be determined ; the expressions 

 so obtained become more complicated. 



Now consider the first of this series of equations 

 logUi^ - logU = log - a + cUo 



