704 ME. A. CAYLEY ON THE EESULTANT OE A SYSTEM OF TWO EQUATIONS. 
the Eesultant is 
{p, q, fja, q, rJJ5, iy.(p, q, rjy, 1)", 
which is equal to 
r" + ^r'(« +/3 + y) 4-^r"(cc" + + /) -\-pqr{a'^ + + /3"y + + ya^ + y"a) + &c. 
Or adopting the notation for symmetric functions used in the memoir above referred to, 
this is I ^3 
{ -\-qi^ ( 1 ) , 
f (2) , 
i +fr (V) , 
f +pqr (21) , 
t +j' (1*) , 
[ +pV (2") , 
1 ( 21 "), 
{ -{-p^q ( 2 " 1 ), 
{ (2") , 
the law of which is best seen by dividing by r" and then writing 
2 = [ 1 ], £=[ 2 ], 
and similarly, 
£ = [!■], ^=[21], &c.; 
the expression would then become 
l + [l](l)+[2](2) + [l^](l^)+[21](21)+[r](P)+[2"](2")+[21"](21") + [2n](2n)+[2^](2^). 
where the terms within the [ ] and ( ) are simply all the partitions of the numbers 
1, 2, 3, 4, 5, 6, the greatest part being 2, and the greatest number of parts bemg 3. 
And in like manner in the general case we have all the partitions of the numbers 
1, 2, 3...«^w, the greatest part being n, and the greatest number of parts being m. 
The symmetric functions (1), (2), (1"), &c. are given in the Tables (b) of the Memoir 
on Symmetric Functions, but it is necessary to remark that in the Tables the fii-st 
coefficient a is put equal to unity, and consequently that there is a power of the 
coefficient a to be restored as a factor: this is at once effected by the condition of 
homogeneity. And it is not by any means necessary to write douii (as for clearness of 
explanation has been done) the preceding expression for the Resultant ; any portion of 
it may be taken out directly from one of the Tables {b). For instance, the bracketed 
( 21 ), 
+ 2 ’ ( 1 *). 
which corresponds to the partitions of the number 3, is to be taken out of tlie Table III(i ). 
as follows : a portion of this Table (consisting as it happens of consecutive lines and 
