MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
207 
X 
X' 
= v. 
x, y, &c. &c. are roots of 
X - v X' = 0, 
which is of the form 
x n — a x n ~ 1 + (b — v) x n ~ 2 -f- &c. = 0, 
where 
S x = a = const., 
and therefore 
S d x = 0 /. S <p d x = <p S d x = 0, 
.*. S J "* p (jji) dx — const. 
I will now add several examples, and I request the reader’s attention to the direct- 
ness with which their solutions are obtained by means of the foregoing- principles. 
In the present paper I have avoided the use of transformations, except that of x = u n , 
because they are unnecessary to the success of the method, and that I am here con- 
sidering general principles rather than individual results. 
§ 6. Examples. 
Ex. 1. Let the proposed integral be 
d. 
/ ' dx 
y/l — x 3 ’ 
This is Mr. Lubbock’s first example in his paper on Abel’s theorem in the Philoso- 
phical Magazine *. The result which he finds is equivalent to this, that if x and y 
satisfy the equation 
xy—{x+y) = 2 , 
then 
f / dx I 3 + f— r- — ■ — g = const. 
J V\ — a? J V 1 — y A 
For the sake of comparison I will take this as the first example of my method, and 
supposing its solution to be unknown, proceed to investigate it as follows : 
= may be put under the form 
d x 
J (1 -x) y/ 1 , + _L 
+ x- 
X 
Pat 
1 + X + x z 
1 — X 
= V 
x 2 + (1 + v) x + (1 — v) = 0, 
x and y must be the roots of this equation -f-. 
* Vol. vi. p. 118. 
f See the note in page 196. 
