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,