118 
ON THE SURFACES THE LOCI OF 
[503 
with the like formula for A pa/3 ; hence 
where 
I a Apa/3 + Ip Apact 
2 *J I a Ip 
= Ax a + By a + Gz a , 
A = —A==.{Ip( 
2 V/.V M 
g a Z-a a W) + I a { cjpZ-apW)}, 
0 = jhj* {h ^^ x +f.Y-e. W) + A (- »X +/oF - o, If)). 
The term in question is thus 
x , 
y > 
Ax a + By a + Gz a , 
æ a) 
Va) 
Za 
Axj) + Byij + Gzfj, 
d'h ) 
yb) 
Zb, . 
Ax c + By c + Gz c> 
X c y 
y c , 
¿c, • 
A Vpda .pd/3 , 
did, 
ya, 
Z d y w d 
viz. replacing the first column by 
— Ax - By 
A s/pda .pd/3 - Ax d - By d - Gz d ; 
this is 
and we have 
= (A x + By) w d . abc ; 
Ax A By = L ^ g«v-fo.y) + I«( g^-fpy)\Z 
2 I a Ip [+ Ip (- - b a y) + I a (- apx - fyy)] IF, 
= 2 i ( - 27 ^- W) - 
if for shortness 
M=(- gpx +fpy) (a a x + b a y) + (- g a x +f a y) (cipx + bpy); 
viz. the whole term is 
w d |- $ Tf| abc. 
Hence the first and second terms together are 
= IT \—z d \/l a Ip + pda. pd/3 —yÆ^. wX abc ; 
I v I0.1 p ) 
viz. this is a multiple of W, which was the theorem to be proved. 
{Surface abcdafi.}