180 
MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
There is, however, one case in which no alteration is produced by this substitution, 
and that is when the variable x is the same function of y that y is of x ; or when 
x = cpy, y — <p x. For then the integral J*y d x or A x d x has the same form with 
fxdy or f<p y dy. 
In this case therefore 
ffdx+ J <py dy = xy + C. 
This equation holds good whether A x d x can be integrated in finite terms, or 
whether it cannot. 
The equations x = <py and y — <p x, manifestly imply that a symmetrical equation 
exists between x and y, and its symmetry is the only requisite condition. In other 
respects it may be any whatever. 
Notwithstanding the simplicity of this reasoning, it does not appear that any ma- 
thematician before Fagnani clearly perceived the important consequences which 
might be deduced from it. But he has obtained from it the following important 
theorem respecting the arcs of the hyperbola. 
If .r be the abscissa of a hyperbola whose principal semi-axis = 1, its arc 
/*. /e 2 ^ s — I 
=J dx Vir=T’ 
where e is the eccentricity, or the distance between the centre and focus. 
Let y be another abscissa, so related to the former that 
e 2 x* — 1 
whence 
e 2 x 2 y 2 = e 2 (x 2 -f- y 2 ) — 1 . 
This equation being symmetrical with respect to x and y, it follows that those letters 
may be permuted, 
ex = 
Multiplying these equations respectively by dx and dy, and then adding them, 
ey d,v + ex dy = dx \f C + dy y/ 
r.exy + C= fdxs/^jrEr+fdy \ZyZTT’ 
which is the theorem in question. 
/ > / t ,« _ | 
dx\/ x'TZTx > 
it may be asked whether the theorem applies to the ellipse, or to the hyperbola, or 
to both curves ? 
