59] 
ON THE THEORY OF ELIMINATION. 
371 
To explain the formation of this final result, write 
V = 
«1, 
ft... 
«2, 
ft 
0.-1... 
ft', 
ft' 
«1 
au! 
ft' 
ft" 
of , a 2 ... 
ft'", ft'" 
which for shortness may be thus represented, 
V = 
12 
12' 
12" 
12"' 
where 12, 12', 12", 12'", f2contain respectively A, A, Z, /, n, n, ... vertical rows, and 
g, k, Ic, m, m, p,... horizontal rows. 
It is obvious, from the form in which these systems have been arranged, what is 
meant by speaking of a certain number of the vertical rows of 12' and the supplementary 
vertical rows of 12; or of a certain number of the horizontal rows of 12" and the 
supplementary horizontal rows of 12', &c. 
Suppose that there is only one set of equations, or g = h: we have here only a 
single system 12, which contains h vertical and h horizontal rows, and V is simply the 
determinant formed with the system of quantities 12. We may write in this case V = Q. 
Suppose that there are two sets of equations, or g — h — k: we have here two 
systems 12, 12', of which 12 contains h vertical and h — k horizontal rows, 12' contains h 
vertical and k horizontal rows. From any k of the h vertical rows of 12' form a 
determinant, and call this Q'; from the supplementary h — k vertical rows of 12 form 
a determinant, and call this Q: then Q' divides Q, and we have V = Q Q'. 
. 47—2