ELIMINANT OF A SET OF GENERAL TERNARY QUADRICS. 675 



and therefore 



M 2 = 



<h x + 2 9i z + 2h iV \v + 2 /i 2 c i 



a 2 x+ 2g 2 z+2h 2 y i 2 y + 2f 2 z c 2 

 a 3 x + 2g 3 z + 2% b 3 y + 2f 3 z c 3 



Now this determinant is what is got by writing the three given quadrics in the 



form 



(a 1 x+2g 1 z + 2h 1 y)x + (6^ + 2/^ + c,z 2 = (K 



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

 (a 3 x+2g 3 z + 2h 3 y)x + (b 3 y + 2f 3 z)y + c 3 z 2 = ()3 



and eliminating dialytically x, y, z 2 . Consequently we have the following simple 

 rule for finding M 2 and its fellow subsidiary quadrics : — 



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

 the terms which have x for a factor, (2) a part containing all the remaining terms 

 which have y for a factor, (3) the "part having z 2 for a factor : and then eliminate 

 dialytically x, y, z 2 . 



(14) From this it is also suggested to obtain other subsidiary quadrics by varying 

 the mode of partitioning the three originals preparatory to eliminating x, y, z 2 . Doing 

 this it will be found that the possible modes are three in number, viz., 



(a 1 x + 2g 1 z + 2h 1 y}x + (& 1 ?/ + 2/ 1 3 )-y + c r z 2 



(a 1 x+2g 1 z }x + (^ + 2/^ + 211^)^ + c v z 2 



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



The first of these, as we have seen, gives 



2 [5]y _ 4[3> 2 + {2[8] + 4[8']}-^ + 2[6}zx + [0}xy , 

 the second gives 



2[9].a 2 - 4[3]z 2 + 2[8}yz + {2[6]-4[6']} zx + [0}xy , 



and the third gives 



2[9].a? + [5]y - 4[3}z 2 + {2[8] + 2[8']} -yz + {2[6]-2[6']}-*b + [0}xy . 



Of these the last may be left out of account, as being half the sum of the two 

 others. The other new form, however, is as simple and as useful as that previously 

 obtained : and, taking the two others derived from it by the cyclical change, we have 

 an alternative form of the eliminant of the 6th order, viz., 



a l 



\ 



<h 



2A 



2<ft 



2\ 



a 2 



h 



C 2 



2/ 2 



2^ 2 



2\ 



a 3 



\ 



H 



2/s 



2<7 3 



2h 3 



■ 



-4[2] 



2[8] 



2[5]-4[5'] 



[0] 



2[7] 



[9] 





-4[3] 



2[8] 



2[6]-4[6'] 



[0] 



-4[1] 2[7] • [0] 2[9] 2[4]-4[4'] 



VOL. XXXIX. PART III. (NO. 26). 5 M 



