374 
Transactions of the Boyal Society of South Africa. 
2. The interest of the matter does not at all lie in the difficulty of 
effecting elimination, but in identifying the peculiar form of eliminant 
referred to by Sylvester, and in ascertaining the process devised for 
establishing it. 
Thus, taking a set of equations illustrating the case where m, 7i = 3, 2, 
namely, 
a^t,u + a^r]U + a^'Cu + a^lv + a^r/v + a^'Cv = 0 ' 
b^^u + b^r]U + b^l^u + b^tv + b^r/v + b^i^v = 0 
Ci^it + C^rjU + C-^'CU + C^lv + C^r}V + Ce'C — vO I 
d^l'U + d^-qu + d^(:,u + d^lv + (^5^/7; + d^^v = 0 
and writing each equation in the form 
{a^u + a^v)^ + (a^w + a^v)ri + ((23?^ + aev)i; = 
we readily obtain four partial resultants like 
a^u-i-a^v a^tt + a^v a^u-^-a^v 
b^u + b^v b^u + b^v b^u -\- b^v 
CjU-\-C^V C^U + C^V C^U-^C(,V 
0, 
or 
I aib^c^\^ur='-{- | \^aJ)^CQ \ + ^ajj^c^ \^ + \aJ)^c^^ ^u'^v 
+ I I a.b^Ce I + I a^b^Ce | + | cifi^c^ \ 1 7iv^ + | aJ)^Ce 1 = 0, 
and thence finally 
\aj)^c^\ \afD^Ce\-\af)jC^ \ + \a^b^c^\ \a,b^Ce\-\aJj^C(,\-{-\a^b^c^\ \afi^Ce\ 
j aj)^d^ I I afj^d(, \ - \ af)^d^ \ + ; ajj^d^ \ \ a^b^d^ \ - \ a^b^d^ | + | a^b^d^ | | aj)^de \ 
I a^cj^ I I a^c^de | - | a^c/l^ \ + 1 a^c^d^ \ | a.c^d^ | - | a^c^de \ + \ a^c^d^ \ \ a^c^de \ 
I b,cj^ I I b.cJe i - I b.c^d^ \ + 1 b^c^d^ \ | Z^.CstZg | - | b^c^de \ + \ b^c^d^ \ \ b^c^de \ 
Here we have first eliminated ^, t}, I, and thereafter 7t3, u'^v, uv^, : 
but we might just as well have eliminated u, v to begin with, and there- 
after l^, rf, 42, ^r], r]^, 4^, the result then being 
a,b. 
a^d^ I 
a^b^\ [a^b^l \a^be\ + \a^b^\ \a,be \ + \ a^b^\ \ a^b^l + la^b^l 
a^c^ I I a^c^ I 1 a^ce \ | a^Ce | + 1 ^365 1 | a,Ce \ + 1 a.^c^ \ | a,c^ \ + \ a^c^ \ 
ajd^ 
hd, 
c^d^ 
0. (B) 
In general the compound determinant in (A) would be of the order 
{m-\-n — l)\jm \ (7^ — 1)!, and the order of the determinants forming its 
elements would be m : similarly in (B) the order of the compound deter- 
