ME. A. CAYLEY’S MEMOIE ON CUEVES OF THE THIED OEDEE. 
431 
Article No. 23. — Com/pletion of the theory in lAouville, and comparison with analogous 
theorems of Hesse. 
In order to convert the foregoing theorem into its reciprocal, we must replace the 
cubic U= 0 by a cmwe of the third class, that is we must consider the coordinates which 
enter into the equation as line coordinates ; and it of course follows that the coordinates 
which enter into the equation PU = 0 must be considered as point coordinates, that is 
we must consider the Pippian as a curve of the third order : we have thus the theorem ; 
The locus of a point such that the tangents drawn from it to three given conics (the first 
or conic poles of any three hues with respect to a curve of the third class) form a pencil 
in involution, is the Pippian considered as a curve of the third order. This in fact 
completes the fundamental theorem in my memoirs in Liouville above referred to, and 
establishes the analogy with Hesse’s results in relation to the Hessian ; to show this 
I set out the two series of theorems as follows : — 
Hesse, in his memoirs “ On Curves of the Third Order and Curves of the Third Class,” 
Crelle, tt. xxviii. xxxvi. and xxxviii., has shown as follows : — 
(a) The locus of a point such that its polars, with respect to the three conics X=0, 
Y=0, Z=0 (or more generally its polars with respect to all the conics of the series 
XX+|«-Y+i'Z = 0) meet in a point, is a cmwe of the thu’d order V=0. 
(j3) Conversely, given a curve of the third order V=: 0, there exists a series of conics 
such that the polars with respect to all the conics of any point whatever of the curve 
V=0, meet in a point. 
(y) The equation of any one of the conics in question is 
, , d\] d\] 
dx dy dz 
= 0 , 
that is, the conic is the first or conic polar of a point (X, //<, u) with respect to a certain curve 
of the third order U=0 ; and this curve is determined by the condition that its Hessian 
is the given curve V=0, that is, we have V=HU. 
(S) The equation V=HU is solved by assuming U=aV+^HV, for we have then 
H(aV+JHV)=AVd-BHV, where A, B are given cubic functions of a, b, and thence 
V=HU=AVd-BHV, or A=l, B=:0 ; the latter equation gives what is alone import- 
ant, the ratio a : b, and it thus appears that there are three distinct series of conics, 
each of them having the above-mentioned relation to the given curve of the third order 
V=0. 
In the memoirs in Liouville above referred to, I have in efiect shown that — 
(«') The locus of a point such that the tangents from it to three conics, represented 
in line coordinates by the equations X=0, Y=0, Z=0 (or more generally with respect 
to any three conics of the series ?X^+^Y+yZ=0) form a pencil in involution, is a curve 
of the third order V=0. 
(/3') Conversely, given a curve of the thu’d order V=0, there exists a series of conics 
such that the tangents from any point whatever of the curve to any three of the conics, 
form a pencil in involution. 
