certain Systems of Equations. 291 



(8.) It is worthy of remark that the same determination may 

 be effected when U and V are both homogeneous cubicj or both 

 homogeneous biquadratic functions. The only difference is that 

 in the former case p is determined by means of a cubic, and in 

 the latter by means of a biquadratic equation. 



(9.) Let us now proceed to the first system given (Ibid. p. 370) 

 by Mr. Sylvester, viz. \ j ;<i^. i:Orjuai>»> 



U=A.a>, V=B.a), W=C.a). 

 We here have 



and, if we make 



X,=A-^B"V, X2=B-'C"V, andW=W'^^ 

 we also have 



A,=BU'-AV^ and A^rrCV'-BW; 



and, if we assume that z=qx, the relations 



Ai=0, and A2=0 {e.) 



will be ordinary simultaneous quadratics in p and q. 



(10.) The solution of these two quadratics would at first sight 

 seem to entail upon us the necessity of solving a biquadratic. 

 This however may be avoided by means of the general theory of 

 linear transformations. Eor, since by linear transformation the 

 system [e.) may, without the occurrence of any equation higher 

 than a cubic, be transformed into two pure quadratics in which 

 the unknowns are linear functions of jo and q, we see that those 

 quantities may be determined (after the transformation) by re- 

 duction and quadratic evolution only. But the above is not the 

 only method of avoiding the occurrence of a biquadratic. The 

 following algebraic artifice enables us to arrive at the same result 

 with perhaps greater ease, simplicity and directness. Valuable 

 and interesting as is the general theory of linear transformation, 

 it may be questionable whether the sphere of its practical use- 

 fulness extends over the pure theory of algebraic equations. 



(11.) Either of the quadratics [e.) may be put under the form* 

 a?2/ + «=0, (/.) 



* For, adopting the notation of my Method of Vanishing Groups (as to 

 which see paragraph XV. et seq. of p. 177 of the last [May] number of the 

 Cambridge and Dublin Mathematical Journal), we have 



provided that 



a?=^i+j^2'v/~~l a^d2/=Ai— ^2a/— !• • • if') 

 If, by means of (/'•)> we determine p and q in terms of x and y, we pass to 

 {g.) by substitution only, and without recourse to any general theory of 

 linear transformation. When A2 is better adapted for our purpose, we may 

 form the function y^2(A2) instead of ySi^O' 



