JiQSS C. A. SCOTT ON PLANE CUEICS. 
2G3 
9 {ij — zf {z - x'f {x — ij'f 
+ 4/3 [iy'z — X {ij + z')] [2zx — y' {z + x')} [2xy' - z {x + y')] - Vlp'x'hjh"^ = 0 . 
Here x, y', z are tlie point coordinates associated with y', C ; we have therefore 
to transform to x, y, z, the original point coordinates. 
Since 
— (‘ 2 X — 9 ) ^ + 2 (3 — A.) &c. 
= (7 — 2 X) ^ +17 + ^, &c., 
the formidie of transformation for x, y\ z (the inverse substitution) can he written 
where 
X = {1 — '1\) X y z, &c. 
V “h 2 (3 — X) X, Ac., 
v = x-i- y z. 
Hence 
y’ ~ z' = 2 {3 - X){y - z), &c. 
and 
ly'z — x' {y' + z) = 2 (3 — X) a {y z — 2x] -f 4 (3 — X)- {2yz — xyy -{■ z)], Ac. 
By means of these, and the value of p in terms of X, the point equation of tlie 
Cayley an is found to be 
108(4- [y - zf {z - xf {x - yf 
+ 4 (4 — \){v{y + 2 — 2a;) + 2 (3 — X) {2yz — zx — xy)} {z, x] [x, y] 
- {u + 2 (3 - \)xY{v + 2(3 - X)y}- [v+ 2 (3 - X)2;}^^= 0. 
The agreement of this with equations (3) and (5) of § 19 may be exhibited by 
writing it in the form 
{y — zf^^ — (9 — 2\fx{x + (4 — X)(y + z)Y{x^ + (2 — X) a: (y + z) + (y + zf] = 0, 
which shows that there is a cusp, tangent to y — z = 0, at the intersection of 
y — z = 0 and a; + (4 — X) (y + 2 ) = 0, i.e., at a; + (4 — X) (1 — a;) = 0, i.e., at 
(X — 3) (a; — 1) -j- 1 = 0 (3); and that the line y — z = 0 also meets the curve on 
X = 0 and on the two lines a;~ + (2 — X) a: (y + 2 ) + (y + 2 )^ = 0 ; i.e., at y — z, 
+ (2 — X) a; (1 — x) + (1 — xf = 0 ; which last reduces to 
\x' — Xx + i = 0 (5).] 
