45^ Mr, Herschel on various points of Analysis, 
mark that the imaginary form here assumed by a is merely 
apparent. 
To proceed. Let the equation (17), multiplied by be inte- 
grated between the limits 0 and and we find after all reduc- 
tions. 
±n 
%rt 
if) 
a» 
(sec 0) ' ' ' . (cos 20) ' ^ . (sec 30)^ ^ ‘ . &c. 
{o+Y, .^ + Y,„_..^}; . . . (25). 
The value of being given by the equation (18) 
Again, if we suppose, for the sake of brevity 
(2) + Y, . 0 " -h . . . Y,„ 0 "" = U ( 0 ) 
log 
— 1 
and^’'L(2).4+Y ,^ + . . . Y,„ . ^ = D"' U(fi), 
we shall obtain, by operating in the same manner on (19) and 
the equations derived from it, expressing the values of the 
series (21) and (22).^ 
erit a — 8 cc. cx bisque valoribus obtiiiebitur 
sec X zz cc 1 — X* + &c.” 
1.2 1.2. 3. 4 
Euler. Inst. Calc. DifF. Pars posterior. Cap. VIII. p. 542. 
The general value of *C(i) as deduced from our equation (24) in terms of the 
numbers ofBERNOuiLLi is as follows; 
/ or \ r — I /„ 2 a’ ,» B zx — 3 /,2jp— 2 
2J;4-1q. ^ 22’+ 1 > ^ (^ — 0 2X-—1 2 (2 1) 
\ t 1.2.3 . . . (2*) ' I 1.2 ... . (20? — 
. B2A? — 3 
1.2.3 
+ 
+ 
.3 . . . (20?) 
, -.0?— I 2.(2“— 1 ) 
( — ly L_ 
-2) 
Bi 
1.2 
1.2 .. . ( 2 PC *""1 
> 
. X 
-f ( — 1) . i . 
* ' ^ 1.2 ... . {zx) 
* Tbe constant added to complete these integrals is determined by makiiyg 0 ~ 0 
•WT 
in wbicb case since cot ( x0) — i, their first members vanish when n is greater 
4 
