260 
ME. W. H. L. BUSSELL ON THE CALCULUS OF SYMBOLS. 
For this purpose put w=l, then (p^”’(^r)=0, and 
V^ — 1^ V^— 1'' 1 ^ 
^(t)) <^)) 
' g’^dTT — \ ' ' g^dir — 1 
g dir — 1 
^W-5 — • 
£^(? 7 r — 1 
We next proceed to show the identity of this value of the coefficient of with that 
formerly obtained. 
The truth of the following theorem is easily seen : — 
(a4_l)/,(T)/2(T)==(£4-l)/,(T)(£4-l}/;(T)+/2(^r)(£^*;-l)/(7r)+/i(T)(£^5^-l)/,(^). 
Hence 
s dir — 1 ) 
+ 
>)ir)^7r}|( ±‘’" )^(,r) [ — l)(^(7r)-i^^(7r))• 
l \ g dir — I / 1 \ ST-^i 
But 
d 
\ £ t/TT — 1 ■' 
ci/TT 
^(7r)£‘^^” ’^dTr^TT ^(tt) .•^— ^ — ^TT ~{- d(7r) . —^ ^TT 
^dir — 1 s^dir — 1 
which agrees with the value of the symbolical coefficient of as before obtained. 
It is proper to add that the same method of investigation applies to all binomials of 
the form of which I have, for the sake of simplicity, selected the case 
I now come to the calculation of the coefficients of the general term of the form of 
the binomial theorem as given in the first memoir. 
