. 1896-97.] Dr Muir on Quaternary Quadrics. 331 
is necessary is a mere change of lettering. Now, whereas in 
Sylvester’s case this letter-changing was fruitful, in the new case 
it is not. The equation (a) here simply reproduces itself ; in other 
words, if is invariant to the cyclical substitution. 
4. Instead of eight equations we have thus only five, and cannot 
therefore proceed. Of course some process of derivation different 
from Sylvester’s may lead to four new equations instead of one, 
but I know as yet of no such process. True, there is one other 
equation outwardly resembling (a), viz., 
CDLa’y - LEA yz + AKBzzo - BGC wx — 0 (/?) ; 
but this, unfortunately, is useless for dialytic purposes when taken 
along with the given set, being obtainable from them by mere 
addition and subtraction of multiples, the operation which gives it 
being 
- CL(1) + LA(2) - AB(3) + BC(4) . 
It corresponds, in fact, to an equation first derived from Sylvester’s 
set in my paper of 1892 (Proc. Boy. Soc. Edin., xx. pp. 300-305). 
It is not unworthy of note in passing, however, that here again 
there is a difference between the analogues. The corresponding 
equation in Sylvester’s case is accompanied by two others. Here 
(fi) is unique, being invariant to the cyclical substitution. 
5. Such being the state of matters, we are apparently forced to 
go afield for a fresh mode of dialytic elimination. One such need 
only be mentioned to be dismissed. 
From each of the five equations, (1), (2), (3), (4), (a), by mul- 
tiplying in succession by w, x, y, z, we can derive four equations, 
and, therefore, can have in all twenty. But the secondary variables 
in these would be 
t; 
y\ 
z 3 , 
x 2 y, 
y 2 *, 
zho , 
w 2 x 
x 2 z , 
y 2 w , 
z 2 x y 
w 2 y 
xhv , 
y 2 %, 
* 2 y, 
W 2 Z 
yzw , 
xzw , 
xyw , 
xyz ; 
and these being also twenty in number, dialytic elimination is 
possible. 
