417 
of Edinburgh, Session 1876-77. 
whence at once the sum of the first n + 1 terms of the expansion of 
(1 - 1)^ is seen to be 
/ _ w P ~ 1 P~ 2 P~n 
y ' 1*2 71 
We obtain merely the same result if we take q v q. 2} &c., as any 
set of consecutive whole numbers ; but from the theorem itself it is 
easy to obtain the equality, 
p f , p + r p + r. p + r + 1 p -f r 
r ( r+l r+l r + 2 r+l 
p + s- 
5=M 
p + r p + r + 1 p + s 
r ' r + 1 s 
Next, write the general identity as follows : — 
If in this we write each letter for its reciprocal we have 
P = <h + ~(P ~ 2i) + J|.(p-$i) {p ~ q 2 ) + &c., 
of which a particular case is the curious formula 
Another is 
1 = cos 0 + cos 26(1 - cos 6) + cos 30{1 - cos 0)(1 - cos 26)+ ... 
+ cos n6( 1 - cos #)...(! - cos (n - 1)(9) 
+ (1 - cos 6)( 1 - cos 26)... (\ - cos n6 ) , 
of which a very interesting case is given by n6 = 2i r . 
