380 me. a. CAYLEY ON THE CONIC OF EIYE-POINTIC 
Proof of the expression for : — 
21. We have 
where 
dK~adx-\-My-\-gclz, 
dQ =hdx + hdy -\-fdz, 
dC = gdx -{-fdy + cdz, 
in which dx, dy^ dz are to be replaced by their values ; we have therefore 
a,H=3„H{ »[(5C-/B>+(/A-AC)fi+(/iB-5A)»] 
-f.[(/C-<;B)X+(«A-jC>+OB-/A>]} 
+ &C., 
where the coefficient of is 
+ C^C — GA)gj^ + (7iB — hA-Y 
+ (2/A — ^B — hC)gjv + (J C — /B) A + (cB —fC^g^ ; 
or, since we have 
{m—l)A~ax-\-hy\-gz^ 
{m — l)B"hx-\-hy -\-fi . 
(m— 1)C —gxYfyYcz , 
the coefficient, omitting the factor which will be afterwards restored, 
OX^ 
+ [ 9{90cA-fyYGz)-G{ax^Jiy-\-9z)']Y 
+ [ h{lix-^hj-\-fz)-h{ax-\-^-\-9^)y 
+ [ %^+/2/ + C2:)-/(/?A’ + %+/2)>X 
+ [ gQix +fy + Gz) -figx+fy + y]kg. 
= Ox" 
+(-3S^+||^K 
-\-(—€x+(B>zY 
Y(—2gx+(By+^z)[Mv 
+(— (§.r+ 
+(— ^^+^y Yg', 
which is equal to 
-(gx"+25|^^"+C^H-24r^/-v+2(gA+2i|X(:.<,>r 
+ (91^+ -f- 
