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395] INVESTIGATIONS IN CONNEXION WITH CASEY’S EQUATION. 
I he proot depended on the assumption, that to the points (f g, K) which lie on 
the curve A, correspond the conics £- -f- + - = 0 which touch the curve 2] this 
x y z 
M. Cremona proves in a very simple manner: the points of J correspond each to each 
with the points of A, or if we please they correspond each to each with the tangents of 
2. To the 6n {n — 1) intersections of J with any curve 12 (viz. \J (fU) + V (gV) + V (hW) =0) 
correspond the 6w(?i-l) common tangents of 2 and the conic f+ ^ + - = 0: if i2 has 
x y z 
a node, two of the 6n (n — 1) intersections coincide, and the corresponding two tangents 
will also coincide, that is i2 having a node (or the point (/ g, h) being on the 
curve A), the conic touches the curve 2. But it is not uninteresting to give an 
independent analytical proof. Write for shortness 
dU = Adx 4- Bdy + Gdz, 
dV — A'dx + B'dy + G'dz, 
dW = A"dx + B"dy + G"dz, 
and let (x, y, z) be the coordinates of a point on /, (X, Y, Z) those of the corre 
sponding point on 2, (/, g, h) those of the corresponding point on A. Write also 
for shortness 
BG'-B'O, CA'-G'A, AB'-A'B = P : Q : R, 
then we have 
AX + BY +CZ = 0, 
AX +B'Y +CZ =0, 
A"X + B" Y + G" Z = 0, 
A \j [u) +B \/ (f) + = °’ 
A' „ +ff „ +0' „ =0, 
A" „ +B" „ +G" „ =0, 
giving 
A , 
B , 
G 
A', 
B', 
G' 
A", 
B", 
G' 
giving 
X : Y : Z=P : 
= 0, which is in fact the equation of the curve J ; and moreover 
Q : R, to determine the point (X, Y, Z) on 2; and 
\/(v) : \/(v) : \/(w) p : Q '■ R ’ 
or, what is the same thing, f : j : h = P’-U : Q‘V : №V, to determine the point 
(f g, h) on A. Treating now (f g, h) as constants, and (Z, F, Z) as current coordinates, 
f + R + -1 = 0, will touch the curve 2 at the point (P, Q, R), if only the 
X Y Z 
the conic