The Besultant of a Set of Homogeneous Lineo-Linear Equations. 377 
whence 
\U23\u-+ {' U26\ + \U53^] uv + \U66\v- = 0. 
Similarly there is obtained 
by eliminating ^, ur], Vrj, I, : and 
\123Q\u-+ ||l536| + i4236| |ztv + |4536N- = 0, 
by eliminating I, rj, u'C, v'C. Elimination of uv, v- from these three 
partial resultants is then all that is required. 
7. Further, this process of ours is of perfectly general application. 
The m + n — 1 given equations being 
\{x^, x^, ?/„ 7/J - 0 
(m + n — 1 rows) 
we eliminate the ^r's in such a way as to obtain a set of 
(m + n-2) !/(w-l)! {n-l)\ 
equations linear and homogeneous in the compound variables 
and then at one stroke eliminate these also. Or, we may begin by elimi- 
nating the y's in such a way as to obtain a set of equations linear and 
homogeneous in the compound variables 
O/j , 0/2, '-iyj^i , 
and then eliminate the x's. 
The number of compound variables in the one set is the same as in the 
other, because the number of combinations of m things taken — l together 
with repetitions allowed is the same as the number of the similar combi- 
nations of n things taken w — 1 together, namely, 
(m + 7i-2)!/(m-l)! (7i-l)! 
Further, it is most important to notice that whether we begin with the 
elimination of the x's or the elimination of the y's there is no real 
