280 Mr. Babbage’s new method of investigating the 
X X 
.c 
y= *-S e~ c fe i£=*dx — % c Je c — l} 
j 2 \ J (1 + £*) 3 J 
These integrals must be taken between the limits x = — oo 
and x = o, putting e*= v this equation is changed into 
v c f - h - v c dv f 
J ii+w)* J 
y 
C f c r V 1 
= —i V C rr V 
2 L J (i+v) s 
I 
~ dv 
where the limits of v are v = o v = l, in the latter integral 
put v = and we have 
but this is equal to c - v c f ■■ v c dv between the limits 
v = o and v = co, which is equal to — 
hence 
. w 
ic sin - 
c 
y — =C'(V— 2*+ &C.) + £*(l*— »*+ &C.) + 
2C sio T + c\ l‘_ 2‘+ &c.) + &c. 
If c= 0-v/ — i we have 
_ = — 6 J (l 2 — 2 2 + &c.) + l 4 — 2 4 + &C. ) — 
0 1 T _"T i — 6 6 ( i 6 — s 6 + &c.) + &c. 
This being added to \ the value given by the theorem (A) 
produces 
t_ * _J 1 l 1 . - , ! I Rr C 
2 r 7T ir ] i +6 l i + i + i + 4 1 ® 4 T 
which is the same value that Euler had assigned to this 
series. 
From the value which has been found forjy, or for the series 
c*(v—~ 2 a + &c.) + r 4 ( i 4 — 2 4 + &c.) + &c. 
I am inclined to conclude that although the series i 2w — 2 2?l + 
