184 MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
Before going further it may be well to adopt, for the sake of brevity, the following 
notation. 
Let there be any number of variables, three for example ; then the sum of their 
wth powers or x n + y n + % n may be briefly written S x 11 , and similarly if f x be any 
function of x, S fx stands for the sum of fx + fy + f z. Also S xy means xy -f x z 
+ y z. And in general 
s / (■*>*/) =f( x ,y) +/&»*) 4 -/(^z) +f(z,x) +f<iy,z) +f{z,y), 
being the sum of all the permutations of the letters. A few examples will render 
this notation familiar. 
Let there be 3 variables, then 
S<z , = ,r + ^ + z 
S xy — xy-\-xz -j- y z 
S x 2 y — x 2 y -f- y 2 x -f- x 2 z + z 2 x -f- y 2 z + z 2 y 
S x 2 y 2 = x 2 y 2 x 2 z 2 + y 2 z 2 . 
Let r = xy z. 
$^=yz+xz + xy = Sxy 
S — = — S xy 
S Jr. = Jn s x "y '> &c - &c - 
Let there be 5 variables, u, v, x, y , z, then 
Swr xy — u v xy -f- w v x z -f- uv y z + u xy z -}- v x y z, & c. &c. &c. 
The greater the number of the variables, the greater is the advantage of this ab 
breviated notation. 
To resume our examples : 
Ex. 5. — Let S x 2 y 2 -f a . xyz + bSxy = c. Then supposing, as before, that 
r + y + z — we find 
2y z — 2 x 2 — ax— b ^(4 c + b 2 ) ■+■ 2 a b x + a 2 x 2 — 4 a x 3 . 
By properly determining the constants a , b, c, this radical may be made to agree 
with any proposed cubic x 3 + a x 2 + (1 x + 7- 
Ex. G. — Let afi -f- y r ' + s 6 = 0, or S x 6 = 0. Here y z or <p x is an implicit func- 
tion of x, only determinable by the solution of the cubic equation 
(m — x 2 ) 3 = ■— x 2 m 2 , 
where m = y z. 
