ME. A. CAYLEY ON THE ANALYTICAL THEOEY OF THE CONIC. 
647 
that is 
mz 
4|x[r(Ar+H;,'+G^')+^'(Hr+B;,'+F^')+^'(Gr+F;?'H-C0'] 
-xr(Ar+Hv+Gr) 
-^r(Hr+B;,'+F^') 
-vr(Gr+Fv+c^'). 
-ny' =g|A(A, ..xr, ri\ ..jr, Z!!^, v)|, 
and similarly 
nn’ -y =g|KA, ..If, H, ?') -^'(A, ..If, f, ?'B, »)}, 
/y -my=i|,{A, ..If, f, ?') -i'(A, ..If, f, ?'I^, 1^, -)}; 
and thence 
(A, H, Gy^nz' —ny', ..)= 
i{(A, H, GB, f., ^.(A, ..If, f, J')’ 
-(A, H, Gif, ?).(A, ..If, f, ?'IX, ,)} 
with the like equations, writing H, B, F and G, F, C in the place of A, H, G succes- 
sively : and we then have 
(A, ..Xmz'—n^, ..f 
=g{(A, .-IX, i^, vjmz'—ny\ ..).(A, ..Xl', »?', Z^f 
1. 
-(A, ..If, ..).(A, ..If, f, ^'IX, i^, r)|. 
But the foregoing values of mz' —ny\ na^ — Iz!, ly'—mod give also 
(A, ,.XX, vXmz!—ny', ,.) 
=k{{A, ..IX, 1^, >y.iA, ..If, f, C'f-[(A, ..IX, »if, f, J')]’}. 
(A, ..If, f, ..) 
= g|(A, ..IX, .If,f, f).(A,..If, f')'-(A, ..If, f, r)’.(A, ..IX, »If, f, f' 
So that 
(A, ..Xwz'— .,)* 
=i(A, ..If, f, S)^{(A, ..IX, ,f.(A, ..If, f, ^')’-[(A, ..IX, 1 ^, .If, f, ?)?) 
= k(A, ..If. f. ?-)*.(«!, ..I.f-K', ..)’• 
4 T 
MDCCCLXIL 
