PHILOSOPHICAL TRANSACTIONS 
I. Researches in the Integral Calculus. — Part II. By II. F. Talbot, Esq. F.R.S. 
Received October 26, — Read November 17, 1836. 
§ i. 
HAVING explained a general method of finding the sums of integrals, I propose to 
apply it to discover the properties of different transcendents, beginning with those of 
the simplest nature. 
In the first place, therefore, I will show its application to the arcs of the circle and 
conic sections. 
As there will be frequent occasion to make use of cubic equations, I shall suppose 
their general form to be 
x 3 — p x 1 q x — r = 0. 
When therefore the letters p q r occur without explanation, it will be understood that 
they represent these coefficients. 
§ 2. Application to the Circle. 
Let us take the integral J and suppose nothing to be previously known con- 
cerning the properties of the function which it represents. 
Let us put, in the first place, 
l x 2 = v x x 2 — ■ v x -f- 1 = 0. 
The two variables x y will be roots of this equation, so that they must satisfy the con- 
dition x y — 1. Also 
d x dy d x d y \ d x 
\ X 2 ' \ + if V X ' V IJ yf X ’ 
dx 
because S — = 0 in any equation whose last term is constant. 
MDCCCXXXVII. 
•/l 
d x 
1 + x 2 
+ 
'J 1 + : 
= const. 
B 
