MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
181 
Let us therefore return to the equation 
whence 
e 2 x 1 y 2 = e 2 ( x 2 + y 2 ) — 1 , 
- (x 2 + y 2 ) + .r 2 ?/ 2 = — Jr 
(1 - x 2 ) (1 -y 2 ) = 1 - Jr- 
In the ellipse the abscissae x, ?/ are necessarily both less than 1, and in the hyperbola 
they are both greater than 1. Therefore in either case the product (1 — x 2 ) (1 — y 2 ) 
is a positive quantity, .*. 1 — must be a positive quantity, which gives 1 > -J, or 
e > 1 . This condition obtains in the hyperbola but not in the ellipse, therefore the 
theorem is not applicable to the latter. An analogous theorem, however, exists for 
the ellipse, which I shall not now stop to examine. 
In imitation of the above proceeding, let us make the more general supposition 
( 71 71 | \ 
e x — 1 \ 
whence 
e n x n y n = e n (x n + y n ) - l, 
a symmetrical equation ; 
( n n , \ 1 
)'■ 
y - 1 7 
Proceeding as before, we find 
exy + C “ dx + f i^~) " dy 
where the notation S J " is employed to express with brevity the sum of two (or any 
number) of similar integrals. 
The sums of many other integrals might be found in the same manner ; but I pro- 
ceed to more general inquiries. 
§3. 
The first idea of a more extended method occurred to me about fifteen years ago, 
when pursuing mathematical studies at Cambridge ; and it was suggested by an at- 
tentive consideration of the process by which Fagnani had rectified the hyperbola, 
as mentioned in the preceding section. The question occurred to me, whether it 
might not be possible to combine three integrals in a similar manner, by supposing 
two symmetrical equations to exist between three variables ? 
