ELIMINANT OF A SET OF GENERAL TERNARY QUADRICS. 



33 



or, since 08' — 56 = —911, into 



(08 + 37 -911>z + (06 + 34-51~2>c + (00 + 310-59)^ = 0. 



From this by cyclical substitution two other equations are obtained, and thence the 



eliminant 



08+ 37-911 06+ 34-512 00 + 310- 59 



00 + lIT- 67 09+ 18-712 04+ 15-610 ( yi ) 



05+ 26-411 00 + 212- 48 07+ 29-810 . 



(36) From the same set of four equations, by the elimination of yz, zx, xy, we find 

 in exactly the same way 



{09'-4'6 + l(8 + 8')}x 2 + {-50'-05' + 26 + 7'(8 + 8')}?/ 2 + {30' + 03 + 68'-6'(8 + 8')}z 2 = 0, 

 and thence the eliminant 



09'-4'6 + l(8 + 8') 



(0 + 0')l + 49'-4'(9 + 9') 



- 04' + l(5-5') + 9'7 



- 05' + 2(6-6') + 7'8 



07' -51 + 2(9 + 9') 



(0 + 0')2 + 57'-5'(7 + 7') 



(0 + 0')3 + 68'-6'(8 + 8') 



- 06' + 3(4-4') + 8'9 



08'-6'5 + 3(7 + 7') 



(y 2 ) 



(37) The obtaining of a set of three equations in x, y, z may be viewed, of course, 

 as the obtaining of a set in x 2 , xy, xz ; or the obtaining of a set in xy, y 2 , zy ; or the 

 obtaining of a set in xz, yz, z 2 . Consequently from the set of four equations in x 2 , y 2 , 

 z 2 , yz, zx, xy which we have been using a three-fold form of result is possible. 



In the first place by the elimination of y 2 , z 2 , yz and subsequent division by x we 

 obtain the equation 



{56 + 37-(8 + 8')0}« + {58' + 32-(8 + 8')5 + 0ll}y + { _53 + 35'-(8 + 8')8 + 6lT}z = 0, 



or, since 56 -08' = 911 and 35 + 88' =611, 



{9lT-37-08}« + {32-58 + 011}?/ + {6lT+35'-88-6ll> = 0, 



and thence the eliminant 



911-37 -08 

 412 + 16' -99 -412 

 21-47 +0l0 



32-58 +011 

 712-18 -09 

 510 + 24' -77 -510 



61 1 + 35' -88 -611 

 13-69+012 

 810-29 -07 



<y») 



In the second place by the elimination of x 2 , z 2 , xz and subsequent division by y we 

 obtain the equation 



{-012 + 69-31}a; + { -512 + 60 + 34}?/ + { -(8 + 8')12 + 66 + 39> = 

 and thence the eliminant 



-012 +69-31 



-(9_+9')10 + 44 + 17' 

 -411 +50 + 26 



-512 +60 + 34 



-010 +47-12 



-(7 + 7')Il + 55 + 28' 



-(8_+8')12 + 66 + 39' 

 -610 +40 + 15 



-011 +58-23 



(yd 



