(578 
Proceedings of the PiOycd Society 
and p, q, r, have the like significations in regard to L. Then from 
the equations— 
M3 - pM 2 + qK - r = 0, 
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 
that is, 
L (M + q) = pM + r, 
pM + r r - pq 
= M + q ’ = P + M + q' 
But we have 
M 3 - pM 2 + </M — r 
(M + q) (ip + m + 9 + 
where 
- 6 = q + p, 
9 = q 2 + <rp + q , 
- co = q 3 + qp 2 + q q + r, 
and thence 
l = nL_j ( M 2 + eM + f') + Pl 
that is, L = &M 2 -f- pM -f- z, where x, p, z are given functions of p, q, r. 
To determine these, observe that 
\/M(M + q) = pM -f r, 
that is 
M 3 - (p 2 - 2q)M 2 + (q 2 - 2pr)M - r 2 = 0, 
which must agree with 
or we have 
whence 
M 3 - pM 2 + qM. - r = 0, 
p 2 - 2q = p, q 2 - 2pr - q, r 2 = r, 
r = V r , 
(q 2 - q)~ = 4 (2q + p>, 
