ELIMINANT OF A SET OF GENERAL TERNARY QUADRICS. 



683 



Instead of this, Professor Nanson, following Sylvester's method, obtains 



— A(mr — qn)-a? — M(ra— cp)-y 3 — R(am — lb)-£ 



+ A(2Mq+pn-lr)-x 2 y + M(2~Rc + bp- qa).y 2 z + R(2Al + nb-cm)-z 2 x 

 + M(2 Ap+qc-br)-xy 2 + R(2M& + cl - na)-yz 2 + A(2Rn+lq-pn)-xz 2 

 + (A + 4AMK)a#». 



Simple, however, though M 3 in this case is as compared with J, there is still a 

 simpler cubic obtainable by subtracting 



from M 3 , viz., the cubic 



where 



Alr-x 2 y 

 Mbr-xy 2 



A = 



Mqy-u-i + Rcz-u 2 + Alx-u 3 



Mqa-y 2 z — Y\.cm-z 2 x 

 TUna-yz 2 — Apm-xz' 1 + (A + A' — amr — AMR)-xyz 



a 

 I 

 P 



b c 

 m n 

 q r 



and A' = 



A b c 

 I M n 

 p q R 



(24) The derived quadrics needed for Salmon's eliminant being differential-quotients 

 of the derived cubic which is needed for Sylvester's eliminant, it is of interest to 

 inquire whether any one of the other cubics of § 20 is such that a differential -quotient 

 of it will vanish for the same set of values as the original quadrics. 



The answer to this may be formulated as follows : — 



If any one of the derived cubics which vanish for the same set of values as the 

 original quadrics have in its determinant form one column composed of the hcdved 

 differential-quotients of the said quadrics, the differential-quotient of the cubic with 

 respect to the same variable will also vanish for that set of values. 



Let the column be the first, thus implying differentiation with respect to x ; and let 

 the cubic in question be 



3mj 



dx 



*dx 



4 to As 



\ Mi 



X, 



fX, 



M 3 



or K say. 



Then, from the mode in which such cubics are obtained, we have 



3w, 



^■ x + \-y + mi- = « 



u„ 



dx 

 i— 2 -x + \-y + ^z 



lf-*-x + \y + iul 2 -z = u z 



VOL. XXXIX. PART III. (NO. 26). 



5 N 



