ME. A. CATLET ON THE ANALYTICAL THEOEY OE THE CONIC. 
655 
28. The eight formulae become all of them the same or very similar for the quadric 
form («, y, which of course implies (A, ..Xl? n, 
Thus selecting any one of them at pleasure, e. g. the formula II. (bis), this becomes 
X { (^^ + + ??'^ + K'^) yyiyl 4- } 
= { ( Z|‘ + mrj + -\-mn -{-n(^')—{lx-{-my-{-nz){^^-\- 
+(r+^'^+r) 
X, y , z 
I, in, n 
where the terms independent of 1''^+;?'^+^'^ destroy each other. Omitting these terms, 
and dividing by the resulting equation is found to be 
1', y , 
x 
2 
r+ 
^x + rly -y^'z , 
|7 -yn'm 
X, y. 
z 
+ W + zK' . 
y+ 
xl •\-ym -|- zn 
1, m. 
n 
lx -^-my -|- nz , 
P-\- 
which is a well-known identical equation. 
Article Nos. 29 to 33, relating to a single conic in connexion with an ineunt or a 
tangent of a conic of double contact. 
29. The formulae assume a very simple form when the point of intersection of the 
two tangents, or the line of junction of the two ineunts of the conic, is an ineunt or a 
tangent of a conic having double contact with the first-mentioned conic. Thus, if to the 
conic 
(a, ..X^, y, zy=0 
tangents are drawn from a point (x’, y' , z') of the conic 
(a, ..Xx, y, zy-y{tx-\-yy-\-lXzJ=^, 
then we have 
{a, ..x^', y, z!j——(Xoi:'-yyy'-yK'^y\ 
and using the form I. (bis), and putting therein (|', V, ^') in the place of the arbitrary 
quantities (?t, gj, v), the equation of the tangent divides out by ^'x' -y'/jX -{•^'z', and omitting 
this factor it becomes 
(a, ..X^, y, z'Xx, y, z)-\-{^x' ^rjy' -\-^'z'){^x-\-yly-\-^'z) 
K'Xy^'—y'z^ zx'—z'x, xy'—x'y)=0, 
which is of the form 
( a , /3 , y X^\ y\ z'X^^ ^)=0, 
i3', y 
i3", / 
4 u 
MDCCCLXIl. 
