196 
MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
or x may be permuted for y. Hence also 
l + x* l + 
*--ir = 9--T L > 
<p being any function. 
Quantities which (like p . - in this example) are not changed in value by 
permuting the roots, maybe termed “ symmetrical functions” of the variables x,y,z, &c. 
or simply “ symmetricals” of the equation whose roots are x, y, z Thus the 
— - — ) is a symmetrical of the equation 
x 2 — v x -f- 1 = 0. 
/ J t 1 — 
But the quantity p ( — — — ) is not a symmetrical of it, because — - — is not equal 
1 — V 3 
t0 ~~y~' 
Ex. 2. Let x, y, z, be roots of x 3 — v x -{- 1 = 0, which may also be written 
y i — vy -\- \ — 0, or z 3 — v z + 1 =0, whence 
l + x 3 i + y 3 i + 5r s 
v = 
X 
y 
whence also 
?>• 
1 + X 3 1 + T/ 3 _ 
X 
<r> • 
y 
= <P 
i + 
1 + X 
Therefore the quantity p . — — — is a symmetrical of the equation X s — v x -f- 1 — 0 : 
1 -j- x^ • 1 j - x^“ } — J— 
but p . — is not a symmetrical of it, because is not equal to — . 
x x y 
x 
These things being premised, it is evident that the same quantity may be a sym- 
metrical of one equation and not so of another. Therefore the problem arises : Any 
quantity being given, to find the equation with respect to which it is symmetrical ? 
] _j_ ^ 1 * 
Ex. 1 . Let — fi— be the given quantity. Put — — = v, v being a general symbol 
for any symmetrical quantity : 
x 2 — v x + 1 =0 
is the required equation, and the indeterminate v is thereby determined to be the sum 
of the roots. 
1 j X*^ 1 | x^ 
Ex. 2. Let — — be the given quantity. Put — - — = v : 
0C X 
.\ x 3 — v x + 1 = 0 
is the required equation, and — v is thereby determined to be the sum of the pro- 
ducts of every two roots *. 
1 | jp j jpi 
* Another example. — Let — — -- — he the given quantity. Put it == v : 
.•. x- + (1 + v) x + (1 — v) = 0 
is the required equation, and (1 + t>) is determined to be the sum of the roots x -f y, and ] — v their pro- 
duct x y. Whence, by eliminating v, we find the following relation between x and y : x y — (x + y) = 2 
This example will be referred to hereafter. 
