460 ON THE ORDER OF CERTAIN SYSTEMS OF ALGEBRAICAL EQUATIONS. [77 
we deduce 
and (ce — d 2 ) (arf + 2bfy + erf) 
+ (cd - be) (brf + 2erf) + drf) 
+ (bd — c 2 ) (erf + 2dtjr] + erf) 
= (ace — ad 2 — b 2 e — c 3 + 2bed) rf, 
ae — 4bd + 3c 2 = 0 , 
ace — ad 2 — b 2 e — c 3 + 2bcd = 0; 
which form the system in question, and may for shortness be represented by 1=0, J= 0. 
The three equations in rf rj may be considered as expressing that 
arf + Sbrfr] + 3crfrf + drf = 0, 
brf + 3crf 2 7) + 3 drf 2 + erf = 0, 
have a pair of equal roots in common; in other words, that it is possible to satisfy 
identically 
(A% + Brj) (arf + 3brfrj + 3c£rf + drf) 4-(A'g + B'rf) (brf + 3erfrj + 3d%rf + erf) = 0. 
Equating to zero the separate terms of this equation, and eliminating A, B, A', B', 
we obtain 
= 0. 
. a , 
3b, 
3c, 
• b , 
3c, 
3d, 
a, 3b, 
3c, 
d , 
b, 3c, 
3d, 
e , 
It is not at first sight obvious what connection these equations have with the 
two, 1=0, J = 0, but by actual expansion they reduce themselves to the following five, 
3 [2 (ce - d 2 ) I — 3eJ] = 0, 
3 [(be — cd) I — 3dJ] — 0, 
[— (ae + 2bd — 3c 2 ) 14- 9c J] = 0, 
3 [(ad — be) I — 3bJ] = 0, 
3 [2 (ac — b 2 ) I — 3aJ] = 0 ; 
which are satisfied by I = 0, J = 0. By the theorem above given, the equations are 
to be considered as forming a system of the tenth order; the system must therefore 
be considered as composed of the system 1 = 0, J =0, and of a system of the fourth 
order. The system of the fourth order may be written in the form 
2 (ac — b 2 ) : ad —be : ae + 2bd — 3c 2 : be — cd : 2 (ce — d 2 ) : 3 J 
3c 
d 
I: