**=A 



100 Mr. A. Cayley on the Porism of 



discriminant of a quadratic function 



aa? + by 2 + cz 2 + 2fyz + Zgzx + 2hxy 

 or (a, b, c,fjff, h){x y y, z) 2 is the determinant 



k=abc-af-bg 2 -ch 2 + 2fyh), 

 and if 



dl 



then the following theorem is true, viz. 



" If (0, oo), (#', oo) are the parameters of the points P, P' in 

 which the conic U=0 is intersected by the tangent, the para- 

 meter of which is p of the conic kU + V=0, then the equations 



Ud=Up-Uk 



U6'=Up-hUk 



determine the parameters 6, 6' of the points in question." And 

 again,— 



" If the variable parameters 0, & are connected by the equation 



110'- 110=211*, 



then the line PP' will be a tangent to the conic kV -f V = 0." 

 Whence, also, — 



" If the sides of a triangle inscribed in the conic U = touch 

 the conies 



*U+V=0 



jfc'U + V=0 



FU + V=0, 

 then the equation 



nk+nk>+nk"=o 



must hold good between the parameters k, k\ W." 



And, conversely, when this equation holds good, there are an 



infinite number of triangles inscribed in the conic U = 0, and the 



sides of which touch the three conies ; and similarly for a polygon 



of any number of sides. 



The algebraical equivalent of the transcendental equation last 



written down is 



= 



let it be required to find what this becomes when £=#=F=0, 

 we have 



4/nk=A + Bk + Ck*+ ..., 



and substituting these values, the determinant divides by 



