442 
MEMOIR ON THE RESULTANT OF A SYSTEM OF TWO EQUATIONS. 
[148 
the expression would then become 
1 + [1] (1) + [2] (2) + [P] (P) + [21] (21) + [P] (P) + [2 2 ] (2 2 ) + [21 2 ] (2P) + [2 2 1] (2 2 1) + [2 3 ] (2 3 ), 
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 being 3. 
And in like manner in the general case we have all the partitions of the numbers 
1, 2, 3,...77171, the greatest part being n, and the greatest number of parts being m. 
The symmetric functions (1), (2), (l 2 ), &c. are given in the Tables (b) of the 
Memoir on Symmetric Functions, but it is necessary to remark that in the Tables 
the first 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 con 
dition of homogeneity. And it is not by any means necessary to write down (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 (6). For instance, 
the bracketed portion 
+ pqr (21), 
+ q 3 (l 3 ) > 
which corresponds to the partitions of the number 3, is to be taken out of the 
Table III (b). as follows: a portion of this Table (consisting as it happens of consecutive 
lines and columns, but this is not in general the case) is 
= 
d 
be 
(21) 
+ 3 
-1 
(I 3 ) 
-1 
if in this we omit the sign =, and in the outside line write for homogeneity ad 
instead of d, and in the outside column, first substituting q, p for 1, 2, then write 
for homogeneity pqr instead of pq, we have 
ad 
be 
pqr 
+ 3 
-1 
q 3 
-1 
viz. pqr x (+ 3ad — 16c) + q 3 (— lad), for the value of the portion in question ; 
equivalent to 
pqr 
q 3 
ad 
+ 3 
-1 
be 
-1 
, or as it may be more conveniently written, 
this is 
in which form it constitutes a part of the expression given in the sequel for the 
Resultant of the two functions in question; and similarly the remainder of the expres 
sion is at once derived from the Tables (6) I. to VI.