Mr, Herschel on various points of Analysis, 457 
S [^+r (7 - t) 1 = D- U (fl) ; (26) 
S{;T^ . log cot(i-^]} = D-'U(,r)-D-'U(^+«);(37) 
-log cot = 
S{ 
^ (2:c--i)^" + 
X |u (tt) + U (S) - U (tt + G) I 
(28). 
In the last of these equations, if we write — [y 
9 j we obtain the following equation, corresponding to (25) 
I \ 2n -j- 1 
(tan 6 ) 
(f) 
I \27Z+I 
. (cot 3^) 
I 1 j 2« + I 
(tan 50) &c = 
log-M 
^D“*U( 7r) + D-~‘ U(— ~-2G)-.D 
— I 
u 
(29). 
These are but a few of the very singular results which may 
be deduced from our equation (12); but I shall forbear to 
extend 'this paper to an unnecessary length by any farther 
applications of it. 
than unity ; but when 7 i — o, the series S | ^ — \ and S f 1 be- 
coming infinite, they take the forms oo x log (i) which is altogether vague and 
inconclusive. Our equations (27) and (28), (29), then, are defective in this case, 
and we can only conclude that the function 
(tan 0)"^ . (cot 3G)'3 . Sec, 
is independent on G, or constant. There seems reason to conclude from other prin- 
ciples however that this constant is t or . more generally, t O 8 i being any 
integer, positive or negative. 
