MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
3 
But S — = 
X 
a _ w , 1 
1 + X 2 X 2 ' X 
~ d x „ d x , 0 d x 
a \ S . , 2 = v& 3 “p o 
1 + X z X X 
: — , and here 
q — 1 and r — — v 
S- = - 
whence 
„ d x d v 
^ 9~ • 
x 1 v * 
and 
~ dx dv 
V X 3 ~ V ’ 
Also 
g dx dv 
X V 
Therefore 
„ d x dv . dv 
aS l +**=“ „ + v=° 
1 + x‘ = C0nst -’ 
which furnishes this other well-known theorem, viz. If three tangents are such that 
the sum of their products = 1, then the sum of the arcs is constant. The constant in 
this case = 90°. 
The same theorem results from the supposition 
i , 
T~, 2 = V + ax; 
1 + x l ' ’ 
for this gives 
x 3 -J- — x 2 -J- x + - — — = 0, 
1 a 1 1 a ’ 
and q — 1 is the only necessary relation between the roots. Also 
But 
S = u S e? <r + aS x d x. 
S x 2 = p 2 - 2 q = ^ - 2 , 
whence 
7 v d v 
k 'i x dx — — i~. 
Also 
c 7 du 
a 
Therefore 
0 dx v dv , lid® ^ ~ _ _ 
S I+^=~ a + „ =°- 
B 2 
Q.E.D. 
