656 
MR. A. CAYLEY ON THE ANALYTICAL THEORY OF THE CONIC. 
where the matrix 
( a , ^ , y ) is = 
/3", y" 
’ ^+5V+^(Gr+F,'+cr), y+f?'-;;^(Hr+B,'+Fr: 
A+5V-^-g(G?+F,'+cf'), j+,'‘ ,/+,'r'+^(Ar+H,'+Gr: 
y+rr+^cnr+Bi'+Fr), /+,'r-^(Ar+H,'+Gf'), c+r- 
30. But instead of further developing these formulae,! prefer to consider the formulae 
which give the points of contact of the tangents in question, \iz. the ineunts of the 
conic (a, .■'Xx, </, zf=0, or the tangents through the point {x\ y\ z') of the conic 
{a, ..Jx, y, zf-\-{^x-\-iiy-\-^'zf=0. 
We have as before 
(a, ..Xx\ y, 
and using the formula IV.(bis) and writing therein (§', r}\ ^') in the place of the arbitrary 
quantities (X, yj, v\ the equation contains the factor +^' 2 ', and dividing by this 
factor, and by K, the line-equation of the ineunt is 
x'^^ry'n^-^'C,■\--=g{x'^ -^-y'n n\ ^'Xl, n, K) 
Selecting the positive sign, the coordinates of the corresponding ineunt are 
1 . 1 f 
+ K +y^' +^'r)( + G^') + ;j'( gx' -\-fy' + cz') — -\-fz') 
y + K(>^^'+i/V+s'?')(H|'-|- B;?'+ F^')+ +hij -\-gz')-^{go(^ -\-fiJ -^cz’) 
and taking (X, Y, Z) for the coordinates of the ineunt in question, and putting for 
shortness 
