2 
MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
We thus obtain a characteristic property of the function 
/ J 1 x - — f x 
namely, that if xy = 1 , 
fx +fy = const. 
The truth of which is otherwise evident, for if x — tan 6, then y = cotan 6 
fx — 6 and fy — 90° — 0. 
.'.fx -f- fy = const. 
Next let us investigate such a relation between three of these integrals that they 
may have an algebraic sum. 
V + X 
Assume 
1 + X- 
a x 
whence 
x 3 + v x 2 + ( 1 — a) x - \- v =. 0, 
where a is any constant quantity. 
The three variables x y z must be roots of this equation, which however gives only 
one necessary relation between them, viz. 
x-\-y-\-z = xyz . 
We have 
1 + X 2 
= 1 
X 
n d X Q d t % n 7 
•. a - — - — s = v & — 4- o d x. 
1 + X z X ' 
But S — = — , and S dx — — dv, 
XV 7 
a S 
d x 
1 + x^ 
d x 
T 
■= dv — dv — 0 , 
0 /* d x 
: . IS / — — 3 = const. 
1 + X 
whence we obtain this well-known theorem in trigonometry. 
If the sum of three tangents equals their product , the sum of the arcs is constant. 
The constant = 180°. 
Next let us suppose 
1 V + X 
1 + X 2 
a x* 
x 3 + (y — a) x 2 + x + v — 0 . 
This gives only one necessary relation between the roots, viz. 
q — xy-\-xz-\-yz=: 1. 
For the two other coefficients ( v — a ) and v, may be made to agree with any two 
arbitrary quantities. Since we have 
