1896 - 97 .] 
Dr Muir on Quaternary Quadrics. 
335 
(4) an equation connecting 
( 5 ) 
( 6 ) 
id) 
( 8 ) 
( 9 ) 
( 10 ) 
, B , C , L , , E , , , H , K ; 
A,B, , ,D, E,F,G,H,K; 
, B,C, ,D,E, F,G,H,K; 
, , C, L, D, E, F, G , H , Iv; 
A , , , L, D, E, F, G,H,K; 
A, ,C, ,D,E,F,G,H,K; 
, B, , L, D, E, F, G,H,K. 
FTow these must be derivable either from a set of six or eight of the 
ten equations cqa 2 = A, etc. For example, the first, — connecting 
A, E, C, D, E, F, — must be derivable from the six in which these 
letters occur, viz., 
cqa .2 = A , a 1 /3 2 + a 2 f3 x = D , 
AA = B , A 72 + &7l = B , 
7lT‘2 = C > 7l a 2 + y-2 a i = F • 
That such is the case is readily seen by primarily eliminating only 
a 2 , /i 2 , y 2 , when there remain the three equations 
Bcq 2 + A/3^ = Da l /3 l 
C/3 1 2 + By 1 2 = E/3 l7l 
Ayj 2 + Ccq 2 = Fyjcq , 
from which, on using Sylvester’s first case,— as must be carefully 
remarked, — we have 
2A D F 
D 2B E 
F E 2C 
as was expected. Similarly, the relation connecting the eight co- 
efficients A, B, , , D, E, F, G, H, K must be derivable from the 
eight equations 
a l a 2 = A , cqyo + a 2 y T = F , 
AA = B , 
a x /? 2 + a 2 /3 1 = D , 
/^i 7-2 B &7l = B > 
£ A + AA = H > 
7 i§ 2 B y 2 8 1 — Iv , 
To make the necessary test we may first eliminate a 2 , /? 2 , the result 
being the six equations 
Ba^ + Aft 2 
= > 
a l 2 ^2 T ASj_ = Gcq , 
A 2 72 + B 7i 
= EA» 
A^ + bs^ha, 
a i 2 72 + A 7 i 
= Faj , 
7iA + 7i S 2 = K - 
