307 
already deduced by Mr. Roberts from entirely different consi- 
derations, and published in Liouville’s Journal de Mathema- 
tiques, May, 1846. . 
Some other interesting results may be obtained from our 
general formule. Thus, if in (2) we put 0 = 0, we will have 
l d. 
Fy MELO ee tig (Eat — te 
from which we may deduce, by an easy transformation, 
; a 44) ind) d 
y Tete Femgy = 4 9¢(Z) 2 trr(k’); (7) 
and, consequently, 
dor 
log (tang)dp __ 1 
\ Vt = Bantgy = #8 (E) FO. © 
If we suppose & to vanish in formule (6) and (7), we ob- 
tain the well-known results, originally given by Euler, 
(" in «Ht 
log (cosp) dg = log (sinp) dp = 37 log 3. 
0 0 
Denoting (1 — sin’) by A, we can also derive from 
_ the above the value of the definite integral 
; - 
\ log (1 = a sin@) dp 
0 yaN 
(9) 
4q For, the sum of the integrals 
iat 
a ip last + 4 sinO) dp bad (. “log (1 (1 — 4 sin@) di 
A 
: ‘may be found from ay, and their difference from the formula 
4 sind dp _ 
S; log i aS 7's Pie we(¥; 9) 
i hich Mr. Roberts has demonstrated in the Journal de Ma- 
pematines May, 1846. 
: 20-2 
