492 Proceedings of Roy cd Society of Edinburgh. [july 18, 
and 2 1 x 1 y 2 \ denote the like sum obtainable from the first two 
sets, then 
2Ky 2 * 3 | = %x.%\y x z 2 \ + %y.%\z x x 2 \ + %z.%\x x y 2 \ (xxxi.) 
This is arrived at by writing out the terms of 2>|?/ 1 z 2 | , of S| 2 pr 2 | , and 
of in parallel columns, thus 
\Vi h\ 
\ Z 1 X 2 I 
1*^1 y^\ 
\Vi h\ 
\h ^ 3 i 
K Vz\ 
Vu-l«n | 
| Z n-lXn 1 
1 • 0 
J— « 
then deriving n results from the members of the first row by 
multiplying by x v y^ z x respectively and adding, multiplying by 
x. 2 , y 2 , z 2 , and adding, and so on ; then treating the second and 
remaining rows in the same way ; and then finally adding all the 
n . \n(n — 1) results together. Each of these results is a vanishing 
or non-vanishing resultant of the 3 rd order, and it will be found 
that each non- vanishing resultant occurs twice with the sign + and 
once with the sign — . 
This process is readily seen to be simply the same as performing 
the multiplications indicated in the right-hand member of (xxxl), i.e ., 
(x 1 + x 2 + . . .+x n )(\y Y z 2 \ + \y^\ +. . .+ \y n -<z n \) 
+ 0/1 + 2/2 + * • .+ 2 /»)( \* X X 2 \ + \Z ± X 3 \ + . . .+ 
+ (z l +z 2 + . . .+z n ) i\x-g/ 2 \ + |aqy 3 j +. . .+ |#»_iy„|) , 
summing every three corresponding terms in the products, and 
writing the sum as a vanishing or non-vanishing resultant. There 
would be n . \n{n- 1) resultants in all; but as each suffix occurs 
n- 1 times in the second factors and once in the first factors, there 
must be in each product n— 1 terms having the said suffix occurring 
twice : consequently there must be n - 1 resultants vanishing on 
account of this recurrence, and therefore altogether n(n - 1) vanish- 
ing resultants. Of the non-vanishing resultants, — in number equal 
to n . \n(ii - 1 ) - n(n ~ 1 ), or \n{n - 1 )(% - 2 ), — each one of the 
form 
I x hVk z i I where h<Jc<l 
