CEETAIN APPLICATIONS TO THE THEORY OF DEFINITE INTEGRALS. 
799 
provided that 
7 
Lt=s 
4s 
a result easily verified. Now let f{ lx) — 
we have 
dx 
C 1 
r ‘ 
ds 
‘sy/ 1+1 
2 
1 ‘ 
\/f6+' 
H) 
Is^ + s'* 
J ^ {l->rX^){l+Px‘^) ^ 
The second member is a function of the same kind as the first, differing only in the 
constants. If we assume s—mt, we can so determine m as to reduce the equation to tlie 
form 
dx C dt 
We shall find 
J 
(1 +Px^) 
i 
( 2 .) 
L=: 
l’ = 
1 _ ^/i-p 
1 + + i 
T+ v/i— 
the relation between x and t being 
t=l'%X\/ A’^ + 1). 
If in the above we make .r=tan(p, ^=tan^, 1 — I — we find 
df 2 f d6 
1 +4 ) \/l — ^2 gjjjSg ’ 
pro\ided that 
J 
-s/l — 4® sin^ip 
(3.) 
, 1 -v/l-F - ^ , , 
h=z 7 ^=^ and tan d—\/ (tan 9 + sec <p). 
1 + \/l-F 
These are of course known relations. 
In the following example, whifch is valuable only for the sake of the principle involved, 
the differential coefficient of a discontinuous function occurs under the sign of integra- 
tion. 
3 
In the same equation (4.), Aid. 38, let /j=0, ^= 2 ^ w=l, A = I; then dropping the 
suffix fi’om the single variable retained, we have 
«' 
wherein ( 7 =« — — and y(( 7 ) vanishes whenever a transcends the limits of x. Suppose 
those limits 0 and 1, and let a be greater than 1. Now a and s increase together, and 
1 ... 1 , , . 1 
5= 
4{a-cry ^=4^’ ^ = 4 («-i) - 
dxf{x) d 
Therefore 
fi 
(5.) 
{a-xf~~^}, 
provided that f(ff) be regarded as a discontinuous function defined in the following 
manner, \iz . — 
