678 



DR THOMAS MUIR ON THE 





X 2 



</ 2 



Z 2 



yz 



zx 



xy 



| Mj b. 2 y + 2/oZ c 3 | 



■f a; 





2[5] 



-4[3] 



2[8] + 4[8] 



2[6] 



[0] 



| Mj c. 2 z + 2g 2 x a 3 | 



* V 



-4[1] 





2[6] 



[0] 



2[9] + 4[9'] 



2[4] 



| Uj a 2 x + 2A 2 y ft 3 1 



-r z 



2[4] 



-4[2] 





2[5] 



[0] 



2[7] + 4[7'] 



If, however, from the eighteen quadrics here tabulated two or more be taken, one at 

 least being taken from the second table, and an aggregate of multiples of these be formed, 

 the resulting quadric, if not symmetrical with regard to the cyclical change, will be 

 one of a triad which also maybe used along with u u u 2 , u s for the purpose of dialytically 

 eliminating x 2 , y 2 , z 2 , yz, zx, xy . In this way, as has been seen, the triad of quadrics 

 which are differential-quotients of the Jacobian may be found, and the triad of § 14. 



Notwithstanding the large number of possible aggregates of the kind here referred 

 to, it does not appear that anything simpler than the triads of §§ 8 and 14 can be found. 



(18) As an illustrative example, let us take Sylvester's special case, where 



u x = Ax 2 + ayz + bzx + cxy \ 

 u 2 = My 2 + lyz +mzx + nxy V 

 u z = B.z 2 +pyz + qzx + rxy ) . 



In accordance with the rule of § 1 3 we write the equations in the form 



(Ax+bz + cy)x + az-y = 



(mz + ny)-x -f (M.y+lz)-y = 



(qz +ry)-x + pz-y + R-z 2 = I 



and eliminating x, y, z 2 we obtain 



Ax + bz + cy 

 mz-\-ny 



az 

 My + lz 



or 



cM-y 2 + (bl — am)-z 2 + (cl — an + bM)-yz 

 Instead of this Sylvester obtained # 



= 0, 



Alzx + AM-xy = 0. 



(ra-cp-cM)y 2 - (b~R + bl-am)-z 2 



+ (aq-bM-bp-cB.-d + a7i)-yz - (AR, + Al)-zx - (Ap + AM)-xy; 



and Nanson, following Salmon's method, obtains t something still more forbidding, viz. :- 



- SA(mr-gn)-x 2 + M(2Ap+gc-br)-y 2 + B.(2Al + bn-mc)-z 2 

 be \ 



+ 4AMR l-yz + 2A(2Rn+lg-pm)-zx + 2A(2Mg +pc-lr)-xy. 



+ 



f\a b c 

 <\l m n 



' P <1 T 



* Sylvester, J. J., " Examples of the Dialytic Method, etc." Cambridge Math. Journ., ii. p. 233. 



t NANSON, E. J., " On the Eliminant of a Set of Quadrics, Ternary or Quaternary," Proc. Roy. Soc. Ediri., xxii. p. 354. 



