MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
201 
1 + x 2 
Multiply each of these equations by the equation — - — = v, and we find 
— S d x = v d v . . (1.) 
i±-f.V^ = o ; ( 2.) 
XX v ’ 
— S x d x — v 2 d v . (3.) 
1 _1_ 
But because is a symmetrical, we may, according to the general principle, 
CC 
introduce it within the sign S. 
S 1 d x — v dv (1.) 
S 1 Jr - 2 X - d x = 0 (2.) 
oc 
S (1 + x 2 ) d x = v 2 d v (3.) 
The integrals of these equations are 
/ I _i_ qr& trft 
— — d x = g- + const (1.) 
/ I “f" X* 
2 ■ d x = const (2.) 
S f{\ -\-x 2 )dx — y + const (3.) 
And we propose to verify these three results by direct integration. First then we have 
f X ~ir~ d x * d V = + log » + logy + const. 
But 
x y = 1 log x + lo gy = 0, 
and 
x 2 y 2 — v 2 — 2. 
,./l±^.^+/i±^^ = |+con S t (1.) 
Secondly we have 
* It is indifferent whether we write S S, and we may remark that the signs may often be per- 
muted. Thus if there are two variables, 
S — Jhdx = x + y 
dSx = Sdx=dx + dy 
-L d S a? 2 = 8 d-^- = x dx + y dy. 
Z z 
2 D 
MDCCCXXXVI. 
