684 



DR THOMAS MUIR ON GENERAL TERNARY QUADRICS. 



and therefore 



xK = 



\ 



Mi 



\ 



M 2 



\ 



Ms 



and therefore, by the theorem regarding the differentiation of a determinant, 



3K 



x tT + K 

 3a: 



3m x 

 dx 



dx 



dx 

 2K 



\ Mi 





\ 2 M-2 



+ 



^3 M 3 





W, ^r- 



+ 



ax, 



dx 

 3X 9 



3X_ 3 

 3 3a; 



3X 2 

 1 3» 



Mi 



Ms 



Ms 





w. 



\ 



+ 



w 2 



X 2 





u 3 



Xs 



+ 



9_Mi 

 3a: 



dx 



g Ms 



3a; 



3w„ 



X * 3^ 



from which it is manifest that any set of values which causes u u u 2 , u 3 , K to vanish will 

 cause 3K/3# to vanish also. 



Had the chosen column been the second, the same mode of reasoning would have 

 sufficed to show that the differential-quotient with respect to y would have vanished ; 

 and similarly in the case of the third column. The theorem is thus established. 



(25) There are seven such cubics among the eighteen of § 20, those of the second and 

 third triads, and the last one of all — the Jacobian, in the case of which the theorem is 

 triply applicable. 



As an example, the first of the third triad may be taken. In its determinant 

 form it is 



a 1 x+g 1 z + 2h 1 y \y+f x z cf+ffl+g-p \ 

 a 2 x+g^+2h 2 y b$+f 2 z c. 2 z+f 2 y+g 2 x j 

 a s x+g a z + 2h0 b$+f s z c 3 z+/0+g 3 x \ , 



and arranged according to powers of the variables it is 



- 2[2}f - [3K 



+ [4}xhj + {2[5]-[5']}-y% + [6]-^ 



+ {[7] + 2[7']}-^ 2 + {[8] + 2[8']}. ?/ z 2 + [9'].*a? 



+ {[0]+2[0']}-^. 



Differentiating with respect to z, on account of the fact that it is the third column of the 

 determinant which is composed of differential-quotients, we have 



[9>* + {2[5]-[5']}y - 3[3K 

 + {2[8] + 4[8']}.yz + 2[6}zx + {[0] + 2[0']}-^ , 



— a quadric which, with the two others obtained from it by the cyclical change, might 

 be used to form an eliminant of the 6th order. In practice, however, it would not be 

 desirable so to use it, as, although it is at least as simple as 3j/3x, it is much more 

 cumbrous than M 2 ; in fact it will be found to be M 2 increased by an aggregate of 

 multiples of the original quadrics, viz., the aggregate l^i./^!- 



