514 Mr. A. Cayley on the Porism of the 



identically with respect to x, y, z, 



(a . . Jx, y, zy><{® . . £/, m > nf-k{lx + my + nzf 



=(«,.. X x » V" g *X x > y> r )x( fl ' • • X**> v * *iX x > y ' r ) 



to a constant factor pres. 



Assume successively x, y, z = &,&, & ; ??, J3, df ; €, tf, C ; 

 it follows that 



: *,*, {33(<a . . J/, »i, nf- (fc/+Bm + dfn) 2 } 

 : *#,{€(« . . J/, m, n) 2 -(<©/ + #>« + Cra) 2 } j 

 or, what is the same thing, 



a? a : y 8 : r s =y 1 ar,(in 9 + cm 2 -2/»m) 

 : 5j a?, (cZ 2 + «n 9 — 2yn/) 

 : x^y^ant 1 + bn 2 —2hlm) . 



It is not necessary for the present purpose, but it may be as 

 well to give the corresponding solution of the problem. 



Given that one of the tangents through the point (£, r\, £) to 

 the conic 



(a, b, c, f, ff, hyx, y, 2) 2 = 



is the line / 1 a? + ?w ] y + » 1 ^ = 0; to find the equation to the other 

 tjuigent. 



Let l 2 x + mvy + ?ic i z = be the other tangent, then 



(« • • J?, V, £f(a . -X x > !/> ~) 2 -(«.-X^^ %Xx,y, zj\ 2 

 = (l x x + m x y + ra j2r) (l^x + m$ + n^z) 



to a constant factor jores. Assume successively y = 0, z=0; 

 z = 0, a = 0; x=0, y = 0, we have 



k ■ »h '■ « 2 =»V'i{«(« • • X%> v > Z) 2 -( a %+ h7 }+9ff} 

 ni l x {b{a..Xtv,KY-{h^ + bv+m % } 

 l x ii h {c{a . . XZ, V, Zf-(ffZ+fv + cl;)*} ; 



or, as they may be more simply written, 



Returning nov. T to the solution of the first problem, I shall 

 for the sake of simplicity consider the formulae obtained by 

 taking for the equation of the conic, 



ctx* + /3y* + yz*=0. 



We see, therefore, that if this conic be intersected by the line 



