INVESTIGATIONS IN CONNEXION WITH CASEY’S EQUATION. 
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equation of the conic is satisfied by these values and by the consecutive values 
P + dP, Q + dQ, R + dR; or what is the same thing, if we have 
that is 
/ 
P 
= 0, 
fdP gdQ hdR 
~P* + Q* + ~R* ~ 
L 
P n - 
9_ 
Q- 
QdR - RdQ : RdP - PdR : PdQ - QdP. 
If the functions on the right-hand side are as U : V : W, then these equations give 
/ : g : h = P*U : Q?V : R^W, 
that is (f g, h) will be a point on the curve A. It is therefore only necessary to 
show that in virtue of the equation J — 0 of the curve J, and of the derived 
equation dJ = 0, we have 
QdR - RdQ : RdP — PdR : PdQ-QdP=U : V : W. 
Take for instance the equation 
V {QdR - RdQ) - U {RdP - PdR) = 0, 
that is 
dR{UP+VQ+ WR) — R {UdP + VdQ + WdR) = 0, 
and this, and the other two equations will be satisfied if only UP + VQ + W r R = 0, 
UdP + VdQ + WdR = 0; we have, neglecting a numerical factor, 
U = Ax + A'y + A"s t 
V = Bx + B'y + B"z, 
W=Cx +C'y + G"z, 
whence, attending to the values of P, Q, R, we have 
UP+VQ+WR = zJ= 0; 
hence also 
UdP + VdQ + WdR + {PdU+ QdV+ RdW) = 0, 
so that 
UdP + VdQ + WdR = 0, 
if only 
PdU+QdV+RdW = 0,