CEETAIN APPLICATIONS TO THE THEOEY OE DEFINITE INTEOEALS. 
789 
may be dispensed with ; a more general conception of the nature of a function supply- 
ing their place. 
36. One remarkable theorem must still be noticed. Since 
sin . 2 ^ 
we have, on taking the logarithms of both sides and differentiating, 
1 . I 
cot.r= — 1 
Hence 
X x-\-'!: X — TT ' x-\-2'n x — 27r 
. Ill 1 
X—COiX = X—- — — — — r-^- 
X x + w x—n x-\-2'k 
&C. 
&C. 
( 1 -) 
Whence, by (2.), art. 32, 
^ dxf{x— cot x) = ^ dvf(v) (2.) 
— 00 C/ —00 
The result may however be generalized. For from (1.), 
-h ? + 
‘ .7J X. TT ' 
«,COt(^— ? 1 ,) = — , . 
'' X — Aj ' X — Ai + tt ‘ X — Aj — It 
m 
Taking the sum of any series of such terms, we shall evidently have 
x—a^ cot {x—'k^ — a^ cot {x—k.^ «„cot {x—k^ 
d -^ dji 
=x- 
X — Aj X — A„ X-t-Tt — Aj x + tt — A„ 
which agrees in form with the function under the sign /'in (2.), art. 32. Hence 
j dxf [x—a^ cot {x—k^) «„cot(^— X„)} = i dvf(v} (3.) 
4 / — 00 4 / — 00 
If we treat in the same way the theorem 
4x^ 
COS^ = ( 1 — — 11 l — 1 — 
we shall arrive at the theorem 
/'* 00 ^oo 
j dxf{x-{-a^txn.{x—\)---\-ay,txn.{x—k,y)}=\ dvf{v). . . . (4.) 
4,-00 4 / — 00 
Essentially, however, this result is involved in (3.), the analogy of which with (2.), 
Art. 32, will be most apparent if we place it in the form 
J* dxflx- 
•'I tan (« — Aj) tan (a?- 
As before, the quantities a^-.a^ must be positive. 
The verification of these theorems by some independent method seems desirable. 
