MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
197 
Ex. 3 . Let x + \/ \ —x 2 be the given quantity. Put #+\/ 1 — a? 2 = v : 
(v — x ) 2 =1 — X 2 
2 x 2 — 2 v x -f- (v 2 — 1) = 0 
v 2 — 1 
X 2 — V X H 2 = 0 
• ^ — • 1 
is the required equation, and v is determined to be the sum of its roots. Also — 3 — 
is equal to their product xy. In other words, the equation must be such, that its 
( CC 7 /) 2 — — 1 
roots x, y, answer the condition x y — . 
Ex. 4 . Let x + v' — be the given quantity. Put x + = v: 
••• ( V ~ f 2 = ~ 
.'. x 3 — 2 v x 2 v 2 x — 1=0 
is the required equation, which is thereby determined to be a cubic, the product of 
whose roots = 1, and v is found to be half the sum of the roots. 
Case of exception . — It is essential to remark, that when the given quantity contains 
only one power of x, it cannot be a symmetrical. Ex. \/ 1 + x n cannot be a sym- 
metrical ; for if it could, we should have \f 1 -{-af = ^/l + y n = \/i + *" = &c., 
whence x — y — z — &c. ; whereas we suppose the roots to be in general all different 
from one another. With this exception the required equation may be easily found 
in most cases by putting the given quantity, or f x = v. And if the roots of the 
equation thus found are denoted by x,y,z it is an immediate consequence 
of the hypothesis that f x — f y = f z = &e. Thus in the last example we have 
* + \f ~7 = y + \Zv = z + Vt- 
Let us now suppose that S dx, the sum of the differentials of the roots, or dx + dy 
+ dz &c., is multiplied by a symmetrical, that is, by one of the above-mentioned 
quantities f x. The product is f x . d x -j- f x . dy + f x .dz-\- See. But in conse- 
quence of the equality fx= f y =fz— & c. the result is the same, if the first term 
is multiplied by f x, the second by fy , the third by fz, and so on. So that the pro- 
duct is f x . d x -j- fy . dy + fz . dz + Sec., which is our abbreviated notation 
= S f x . d x. 
.'. f x . S d x = Sfx . d x ( I .) 
This theorem is of the greatest importance, and will be of constant use in the sequel. 
It must not be forgotten that it is only true when f x is a symmetrical, and therefore 
capable of being represented by v. Replacing/\r by v , it becomes v Sdx = Sr dx. 
In this form it is self-evident, because v remains the same, however the letters 
• r > y> z, . . . . are permuted. 
