MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
203 
The integral of this last equation is 
s/VMr^**- h*+ c > 
which means that 
[• ■ d x +f\/ff ■ dt J = ■§■ (* + yf + c > 
provided that x y — 1 . For since x, y, are roots of x 2 — v x + 1 = 0, x + y — v, a 
variable quantity, and xy — 1 is the only condition which the variables must satisfy. 
Now assume x — u 2 , y — t 2 , and the relation between the new variables will be 
u 2 t 2 — 1, or u t — 1 ; and equation (1.) becomes, when divided by 2, 
— ( u 2 + t 2 )^ + C = 1 -f- m 4 . d u -f- 1 -| - ft . dt, 
whence the following theorem. 
If u, t, two ordinates of the cubic parabola*, are reciprocals, so that ut = 1, then the 
1 3 
sum of the two corresponding arcs of the curve = y (u 2 + t 2 )~ + const. 
The reader may wish to see this result also verified by direct integration. Since 
then ut — 1, let us write ~ instead of t in the equation 
y (u 2 -{- + C — f 1 + m 4 . d u -j- fsj 1 ft . dt, 
and it becomes 
y ( u 2 + ^) T + C —j'tj 1 + w 4 . du — f 1 + ui • 
or 
’ +C =f. Jl+u'.du^l - f), 
which ought to be identically true, whatever be the value of u. To see that it is so 
in reality, we have only to differentiate the first part of the equation, and we find its 
differential to be 
2v/l + w 4 . du — ^ 1 U ^ . d u 
— J\ + m 4 . du (2 — — y ?< -) = */l + m 4 du (l — ^), 
which is the differential of the second part of the equation. 
Let us now show the application of the method to formulae containing cubic ra- 
dicals. 
* The equation to the cubic parabola, whose coordinates are u, u', being 3 vl — u 3 , it follows that the arc 
= J* iu */l + u*. 
2 d 2 
