458 Prof. Sylvester on the Theorems 



given ones, viz. 



(bu -f Uv -\-b"w)y—(au + a!v + a! r iv) x = 0, 

 (cu + c'v + c n w)y— (bu + b'v + b n w)x = 0, 

 (du -f d!v + d"w)y— (cu -f c'v -f c" w)# = 0, 

 (eu + e'v + e"w)y— (du + ^'v -f d ,! w)x=0. 



These equations may be satisfied by making simultaneously 



u=0, v = 0, iv = 0, 



all of which (since u, v, w are minors of the same rectangular 

 matrix) may exist simultaneously, provided 



bed 



b' 6 d' =0. 



b" c" d" 



Rejecting (as before) this irrelevant factor, it remains to find the 

 resultant of the system of equations in x, y ; u, v, w, above written, 

 defined as the characteristic of the possibility of their coexistence 

 for some particular system of values of a?, y; u, v, zv, but with joint 

 and several exclusion of the system a?=0, y = 0, and of the system 

 u = 0, v — 0, w — 0. 



So, in like manner, in the general case we shall obtain a similar 

 system of (m + 1) homogeneous equations linear in x, y, and also 

 in u 1} u q ,. . . u m ; and R will be the resultant of this system, sub- 

 ject to the same condition as to the exclusion of zero systems of 

 x, y, and of u v u 2 , . . . u m as in the particular instance above 

 treated. Such a resultant, as hinted at the outset, is entitled to 

 the name of a double determinant. In general a double deter- 

 minant will refer to two systems of variables, one p, the other q 

 in number, and to (p + q — 1) equations between them. 



In the particular instance before us, one of these quantities, 

 say q, is the number 2. There is, moreover, a further particu- 

 larity (but which as it happens does not at all influence the form 

 of the solution), consisting in the fact that the equations are of 

 the recurring form 



I'M 



—\i Q oc=§, 



Wj 



— Lj x=0, 



L s2/ 



— L 2 #=0, 



L p+l .y — L p x = 0, 



where L , L„ . . . Lp +1 are each of them linear homogeneous 

 functions of u v u 2 , . . . u p . This gives rise to an identification of 

 the resultants of two matrices of very different appearance — one 



