678 
Proceedings of the Royal Society 
and p, q, r, have the like significations in regard to L. Then from 
the equations — 
M 3 — pM 2 + - r Ho, 
L 3 - pL 2 + qL - r = 0 , 
L 2 - M, 
we can express L as a linear function of M 2 , M, M°, with co- 
efficients depending on p, q, r ; and obtain expressions for p, q, r, 
in terms of p , q, r. 
We have 
L (M + q) - pM + r , 
that is, 
L 
pM + r r - pq 
M + q ’ = P + M + q' 
But we have 
M 3 - pM 2 + - r = (M + q) (l’ + «M +P+ HT- q ) 
where 
- 0:= q + p, 
<p = q 2 + q£> + q , 
- (O = q 3 + qp 2 + qgr + r , 
and thence 
L = pq m r (M^ + 6M + <p) + p, 
that is, L = £cM 2 + 2/M + z, where x, y y z are given functions of p, q, r. 
To determine these, observe that 
that is 
\/M(M + q) = pM -f r, 
M 3 - (p 2 - 2q)M 2 +' (q 2 - 2pr)M - r 2 = 0 , 
which must agree with 
M 3 - pM 2 + ff M-r=0, 
or we have 
p 2 - 2q = p, q 2 - 2pr = q, r 2 = r, 
r = «/r, 
(q 2 - if = i (2q + p)r , 
whence 
