in elimination between Quadratic Functions. 217 



functions, and one linear function of x 9 y 9 ss 9 t 9 contains the 

 factor 



□ □ {^U + j*V + (te + »iy + »*+p/) tt }. 



X[i xyzt 



This last quantity is of the 4 x 3th, i. e. the 12th order in 

 respect of the coefficients in U and V combined ; of the 4 x 2th, 

 i. e. the 8th order in respect of /, m, n 9 p ; and of the zero 

 order in respect of q 9 r, s 9 t„ 



The resultant which contains it is of the (4 + 4-f 2.4)th, 

 i. e, 16th order in respect to the coefficients in U and V; of 

 the (4 -f 8)th, i. e. the 12th, in respect of /, m, n 9 p ; and of the 

 4th in respect of q 9 r 9 s, t. Hence the special (and, as far as 

 the geometry of the question is concerned, the unnecessary, I 

 may not say extraneous or irrelevant) factor which enters into 

 the resultant is of the 4th order in respect to the combined 

 coefficients of U and V* ; and of the same order in respect to 

 l 9 m, n 9 p 9 and in respect to q, r, s 9 t. 



I have not yet succeeded in divining its general value. 



In the very particular example, of the system, 



a# 2 -f /3j/ 2 =0 



cz*+dfi=0 



Ix + my -f- nz +pt = 



ax fy 



cz dt 



1 m n p 

 q OJ 



I find that the double determinant is 



c 2 d 2 ot?P 2 {cp 2 + dn 2 )*(m*cc + l q (Sf, 



and the resultant is 



? 4 c 2 c?V/3 4 (^ 2 + ^ 2 ) 4 . (m 2 a + Z 2 0) 2 , 



giving as the special factor 



q\^(cp^ + dn 2 )\ 



I believe that the theorem which I have here given for de- 

 termining the condition that lx-\-my + nz+pt shall be a tan- 

 gent plane to the intersection of two conoids U and V, viz. 

 that the determinant of 



xU + V+{tx + my + nz +pt)u 



shall have two equal roots, is altogether novel. 



* And consequently of the second in respect to the separate coefficients 

 of each. 



= 0, 



