ELIMINANT OF A SET OF GENERAL TERNARY QUADRICS. 



679 



(19) Leaving now the eliminant of the 6th order let us return to the problem of 

 finding the cubic, M 3 , to be used in place of the Jacobian in forming the eliminant of 

 the 10th order. 



As we have seen 



M 3 = M 2 -z + 21^6^3^ + 4|m 1 / 2 5t 8 |-2. 



Now the sum of the last two terms on the right is 



and from § 13 



Consequently we have 



= u, 



M 2 -z 



\y 2# 3 1 + | % x 



\y + 2U 2# 3 |; 

 = I u t b 2 y + 2f 2 z 



2/ 2 * *9, 



3 I ' 



Go — 



= U-, 



M Q = 



hv+ 2 A z h zx ~ 



\y + 1U c 3 zx- 1 + 2g 3 



Now diminishing each element of the first column by y times the corresponding 

 element of the second column and by zx times the corresponding element of the third 

 column we obtain 



and thence 



M 3 = | a 1 x 2 + 2h 1 xy hy + Z/z* G l zx~ x -\-2g % 



M 9 = 



a 1 x+2h 1 y 



\y+iU 



c 1 z+2g 1 x 



a 2 x + 2h 2 y 



b 2 y+%U 



c 2 z+2g 2 x 



a z x+27i i y 



t>0+2fe 



c z z + 2g 3 x 



But this determinant is what is got by writing the three given equations in the form 



(a 1 x + 2h 1 y)-x + (b 1 y+2f 1 z)-y + (c l z + 2g 1 x)-z = 

 (a 2 x + 2h 2 y)-x + (b 2 y + 2f 2 z)-y + (c 2 z+2g 2 x)-z = 

 (a 3 x+2h 3 y)-x + (b. d y + 2/ z}y + (c s z+2g z x)-z = 



and eliminating dialytically x, y, z. Consequently we have the following simple rule 

 for finding M 3 : — 



Separate each of the given quadrics into three parts, viz., (I) a part containing the 

 terms which have x for a factor and do not involve z, (2) a part containing the terms 

 which have y for a factor and do not involve x, (3) a part containing the terms which 

 have zfor a factor and do not involve y : and then eliminate dialytically x, j, z. 



(20) It is thus seen that M 3 and J resemble each other in being both obtainable by 

 dialytically eliminating the first powers of x, y, z from the three given quadrics, just as 

 M 2 and its fellow resemble each other in being obtainable by the elimination of x, y, z\ 



The other cubics of the family to which M 3 and J belong arise from different modes 

 of partitioning a general quadric into three terms having x, y, z respectively for factors ; 

 and on trial it will be found that these modes are eighteen in number, viz., 



