452] 
261 
452. 
ON AN ANALYTICAL THEOREM FROM A NEW POINT OF VIEW. 
[From the Proceedings of the London Mathematical Society, vol. hi. (1869—1871), 
pp. 220, 221. Read February 9, 1871.] 
The theorem is a well-known one, derived from the equation 
(az- + 2bz + c)w 2 -f 2 (a'z 2 + 2b'z + c') iv 4- aV + 2b"z + c"= 0 ; 
viz., considering this equation as establishing a relation between the variables z and w, 
and writing it in the forms 
2u = Aw 2 + 2 Bw + C = A'z 2 + 2 B'z + C = 0, 
(where, of course, A, B, C are quadric functions of z, and A', B’, C quadric functions 
of w,) we have 
0 = ~ dio + ^ dz, = (Aw + B) dw + (A'z -1- B') dz\ 
dwdz 
but in virtue of the equation u = 0, we have Aiv + R = VR 2 — AC, and A'z + B' = \f B' 2 — Á'C', 
and the differential equation thus becomes 
dw + dz _ 0 
VR' 2 - A'G' \/R- - A G ~ ’ 
where B'-—A'C and B 2 — AC are quartic functions of w and 0 respectively. This is, 
of course, integrable (viz., the integral is the original equation u = 0); and it follows, 
from the theory of elliptic functions, that the two quartic functions must be linearly 
transformable into each other; viz., they must have the same absolute invariant P -í- J 2 . 
It is, in fact, easy to verify, not only that this is so, but that the two functions 
have the same quadrinvariant I, and the same cubinvariant J.