388 
ME. A. CATLET OX THE COXIC OE ERTl-POIXTIC 
suggested by a geometrical theorem of Mr. Salmon’s. In fact the tangent at the point 
[xX, z) meets the cubic in the tangential of this point, and the coordinates of the 
tangential are z{x^~ify Calhng these r, the equation of the 
tangent to the cubic at the tangential is 
(r+2/p3^)X+(>J^+2^ri)Y+(r+2/|?j)Z = 0. 
Now we have identically, 
3^yz"|(3ZV” (1 +2ry)XH- (3^y — (1 +2Z%2)Y+ (3ZV— (1 + 2P>t)z} 
— Q { + 2 /p)X + (^^ + 2 Izxyi -\-2lxyyL] 
= {(f+2/;3^)X+(;?'*+2/^|)Y+(r+2/|;?)Z} 
j^j]{x\-y^-z^^2lyz)K+y\-z^-x^+2lzx)Y-^z^{-x^-f+^lTy)Zh 
In fact this relation will be true if only 
^xYz^{^Vx‘^ - ( I + 2 P) 2 /z) - -4- 2 lyz) 
=r.U^4-3/^-z=’+2^j/z). 
And substituting for |, ^ and Q their values, the left-hand side is 
3^Y2;^(3ZV-(l+2r)^^) 
- (^^ + 2 hjz)Yz^^z^x^-^aff - 3r^aY^') 
—x\y^—zY 
--2hjz{z^ —x^){x^ —y ^) ; 
and expanding and reducing, the result is 
{if-\-zY - + 2^f( - 2^'y + A + + 12 /vry^'- ; 
whence, dividing by x‘^, the equation becomes 
_(y»+ 2 »)(®«+)/»+j»)+ 2 ?a.-j/ 2 (ar‘— 2(i/*+s’))+12fV/j’ 
=(-2/»-z»+2;y3)(a-*+2/’+-’+6/.ryj), 
which is identically true. 
36. Hence in the identical equation putting U = 0, we see that the equation ot the 
five-pointic conic may be written 
9{l-\-d>l^)xyz%x, y, z, lx, ly, IzJA. Y, Z)^ 
q_ { Y+2/2/5)X+Y+2fea’)Y+Y+2/YZ} x 
{(f +2?;70X+ Y +2/^1 )Y+(r+2/?^ )Z} 
= 0 , 
where |, n, I stand for x{f-z% y{z^-xY z{x^-f), the coordinates of the tangential of 
the given point, and which puts in evidence the geometrical theorem above referred to. 
viz. ... • • i. 
Theorem. The common chord of the five-pointic conic and the polar conic is the 
tangent to the cubic at the tangential of the given point. 
