PEOFESSOE CAYLEY ON CUEVATUEE AND OETIIOGONAL SUEFACES. 
251 
the difference is found to be 
= «{(A, . -IX, Y, Z) 2 +AY 2 } 
+ b-\(A, . .JX, Y, Z) 2 +BY 2 } 
+^(A,..XX, Y, Z) 2 + CV 2 } 
+2/{(A+B+C)YZ + FV 2 } 
+2y{(A+B+C)ZX + GV 2 } 
+ 2A{(A+B+C)XY+HY 2 }, 
which, on account of (A, . -XX, Y, Z) 2 =0, and A+B + C=0, reduces itself to 
(A, . .Ja, . . .) . V 2 . 
49. We have 
A«+H/i+Gy= «(2AZ — 2yY) 
+A( yX- fY-(a-b) Z) 
+R/Z- hX-(c-ct) Y) 
= X(% -%) 
+ Y («y- ga-(ga+fh+cg)) 
+Z (A«— «// + (//«+ M+/y)) ’ 
or observing that in the coefficients of Y and Z the second terms each vanish, this is 
A a + HA + Gy =X(Ay — gh ) + Y(ga— ag ) + Z («A — A«), and similarly 
H/7+ B b+Ff=X(bf-fb ) +Y(fh -hf)+Y(hb -A A), 
Gy +11 A + F c=X(fc-cf)+Y(cg -g C )+Z(gf-Jg). 
Adding these equations, the coefficient of X is the difference of two expressions each of 
which vanishes ; and the like as regards the coefficients of Y and Z ; that is, we have 
and consequently 
AX, 
BY, 
VL 
x, 
Y, 
Z 
YX, 
lY, 
U 
the required relation. 
(A, ..+«, ..) = 0; 
= ((A), . .Ja, . . ,) = -(A, . . .JtX, iY, iZf, 
