ELIMINANT OF A SET OF GENERAL TERNARY QUADRICS. 395 
47’ 48 44’ l 17’ 19 ll’ 
Dabs alt oe lene Uae 6, (ekg: 
59 58’ ; 5 55’ * 28’ Qi 22! 
Te) ie Sr et ee a ee ee RP oe 
67 69 66’ 38 39’ 33’ 
9 7 O+ = Cece T 9 One ae aan 
6-1’ AL 9M 5) = 3) ; | = dl Bes By 1-6 
This is a verification of the identity 
B(4-2)A, = 3(2-4)A, 
already obtained in § 48. The latter form of it shows that both of the four-line deter- 
minants are invariant to the cyclical substitution; and as the interchange of § 42 
transforms the one into the other, it follows that both are doubly-invariant. 
(50) The two new forms of eliminant just reached make clear the fact that if one 
of the four sets of determinants 
Pease ee Orsor of) gio) oe Sos 
vanishes, the eliminant takes the form of a single four-line determinant. 
For example, 
if 4, 5, 6 have each the value zero, the eliminant is 
| qc 19 > es 
esi A ae a aaa, | 
| 
| 28° OT 22! 
8 ore pete oe 
ae 3 3 
38 ee 
eed 9 == = 
; 0+5 
4 ry 6) 00" |) 
We are thus brought to consider the problem of finding the set of four equations whose 
coefficients are the elements of this determinant. In the quest for a solution we are 
not without a lead, since for one of the very special cases brought into notice by 
SYLVESTER the desired set of equations has already been obtained.* 
(51) From the fundamental set of equations there can be deduced (§ 33) 
[| wieotg | = —9a?+2y2-Tyzt+lay = 0, 
| wad, | = O22 +lyzt+4ea+Tay = 0, 
Ill 
and from these by multiplication by z and y respectively we obtain two equations in- 
volving the desirable facients yz*, zx?, xy”, xyz, together with the undesirable y’z. On 
eliminating the last mentioned there results 
(02 + 17’)yz? + 1920? + 2Txy? + (24 -1))ayz = 0, 
and by cyclical substitution 
38yz? + (03 + 28’)za? + 2Tary? + (35 — 22’ \ayz 
— 3832? + 1920? + (01 + 39’ )ay? + (16 — 33')ayz 
= 0, 
OF 
* Proc. Roy. Soc. Hdin., xx. p. 377. 
TRANS. ROY. SOC. EDIN., VOL. XLI. PART II. (NO. 16). 58 
