GG8 DR THOMAS MUIR ON THE 



(4) At first sight this might appear to be a generalisation of Sylvester's three 

 special cases, but a little closer inspection suffices to show that the subsidiary quadrics 

 obtained by differentiating the Jacobian are quite different in form from those to 

 which Sylvester was led. Thus, to take the simplest of the special cases, when 



w, = Af-2C'xy+Bx*- 

 u 2 = B2 2 -2A>+Cy 

 u 3 = Cx*-2B'zx+Az 2 



we readily find 



J = (Bx-C'y)(Cy-A'z)(Az-B'x) + (Bz-A'y) (Gx-B'z) (Ay-C'x), 



and 



y = C(C'B'-AA> 2 + B(B'C'-AA> 2 



+ 2(ABC-A'B'C>z + 2B(A'B'-CC>; + 2C(G'A'-BB')xi/, 



~ = A(A'C'-BB> 2 + .. . 

 ^ J = B(B'A'-CC> 2 + ... 



Now instead of these lengthy and complicated subsidiary quadrics those which 

 Sylvester used were 



A'* 2 + Ayz — B'xy — C'zx, 



and two others like it. # 



An equally marked discrepancy is observed when the like examination is made 

 of the two other special cases, the advantage in all three being on Sylvester's side. 

 An inquiry is thus suggested as to whether there be not a general method, simpler 

 than that given by Salmon, and leading in the special cases mentioned to Sylvester's 

 results. 



(5) The probability of this question being answerable in the affirmative seemed to 

 be increased by a consideration of the results obtained in a recent investigation t of 

 the first two cases ; for it was then found that not only was there a considerable 

 variety of suitable subsidiary quadrics available, but that in the second case those used 

 by Sylvester, comparatively simple though they were, were far from being the 

 simplest possible. 



The main object of the present paper is to settle the point thus raised. 



* The connection between 3J/?a; and the corresponding quadric u?ed by Sylvester is 



~ c + CA'u l - B'C'm 2 + A'Bm 3 = 2BC { A'* 2 + Ayz -B'xy -C'zx}. 



t Mum, T., " A Problem of Sylvester's in Elimination," Proc. Roy. Soc. Edin., xx. pp. 300-305 ; Caylet, A., 

 ■ Note on Dr Muirt paper, ' A Problem, etc.,'" ibid., xx. pp. 306-308 ; Mum, T., "Further Note on 'A Problem, eto,,'" 

 ibid., xx. pp. 371 -382 ; Muir, T., " On the Eliminant of a Set of Ternary Quadrics," ibid., xxi. pp. 220-234. 



