22 Mr. A. Cayley on the Porism of the 



Now 



{mn'-m'ny= (m^ + n^) (m'^ + n''^) - {mm' + mi'f 



=zlH'^-{mm' + nn'f 

 = — {W + mm! + nn') { — ll' + mm' + «"') ; 

 and making the like change in the analogous expressions, and 

 putting for shortness 



— bc + ca + ab = a 

 bc—ca-{-ab = /3 

 bc + ca — ab=y, 

 the conditions in question become 



{I'l" + m'm" + n'«") («Z7" + fim'm!' + yn'n") = 



(/"/ + 77i"}n+n"n) {al"l + /3m"m + yn"n) =0 



(//' + mm' -f nn') {uW + /3mm' + ynn') = 0. 



The proper solution is that given by the system of equations 



P +m^ +n'^ =0 



al!l" + ^m'm"-{-yn'n"=0 



aU'l +0m"m +yn"n = 



all' + |S?»»i' +ynn' =0. 



f f f" a 



And by writing 1= -^, l'= -^, /"= -^, m= — ^, &c., 



Va V« V« V/3 



A = -, B = ^, C = -, the^e equations become 

 a P 7 



A/2 +B/ +C/r =0 

 A/''^+B/2 4-CA'2=0 

 A/"HB/2 + CA"2 = 0, 



//'+// + A'A"=0 

 ff+9''9+h"h=Q 

 ff, +gg< +hh< =0. 



The first of which systems expresses that the points (/, g, h), 

 if) y') ^')> if") 9") " ') ^^'^ points in the conic 

 Aa'2 + B/ + C^2^0; 



and the second condition expresses that each of the points in 

 question is the pole with respect to the conic 



a:2 + y2 + ^2 = o 



