MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
185 
Notwithstanding the complicated nature of the function p x, we still have 
S/p x . dx — xy z C. 
But the most interesting result was obtained by combining the equations 
x + y + z = 0 
x y -j- x % + y % = 
x-y* z 2 
or, in the compendious notation, putting xy z — r, 
— p 2 
S x = 0 S xy = — — : 
2 
*/ 1 + x 4 
~ 2 = <P 
whence 
yz = 
and by the general theorem A 
xy z + C = S^/*p x . dx (1.) 
/ /> ^ ^ iji 2 
p x .dx consists of two parts, of which the latter is J — — = ~. There- 
2,2,2 0 1 2Siw — r 
L — — 9 Si =£ — • ci 
fore the sum of three such portions = —+ — + — = 2S— = 
A x y z x 
by hypothesis S x y — 4 
Hence if we put 
equation (1.) becomes 
whence 
r 2 
since 
J' f - X . d x = J* p x . dx, 
r -f- C = 2 s/* p x .dx } — ; 
S J* p x . d x — const. 
Now this result is highly deserving of attention ; for the integral which we have 
here called J ' p x . d x — J * V - 1 - / x .dx, is no other than the arc of an equilateral 
hyperbola whose abscissa is x, the equation of the curve being referred to its asym- 
ptotes. When I arrived at this result, I immediately perceived that (provided there 
were no error in the reasoning, of which I at first entertained some doubts,) it was 
an entirely new and undiscovered property of the hyperbola. I therefore proceeded 
to verify it by calculating numerical examples. 
The theorem may be stated thus : If three abscissae of an equilateral hyperbola 
y.2 
verify the equations S .r = 0, S xy — — -p , the sum of the arcs subtended by those 
abscissae = f r + const. In order to eliminate the constant, the value of which was 
unknown, I supposed one of the abscissae x to assume some other value x\ and there- 
mdcccxxxvi. 2 B 
