432 
ME. A. CAYLEY’S MEMOIE OX CUEYES OF THE THIEH OEHEE. 
Now, considering the coordinates which enter into the equation of the Pippian as point 
coordinates, and consequently the Pippian as a cuiwe of the thu'd order, I am able to 
add as follows : — 
(y') The equation in line coordinates of any one of the conics in question is 
d\J. dV dJJ_ 
dri 
dK, 
that is, the conic is the first or conic polar of a line (A, [Jb, v) \^ith respect to a certain cun e 
of the third class U = 0 ; and this curve is determined by the condition that its Pippian 
is the given curve of the third order V=0, that is, we have V=PU. 
(^ ) The equation V=PU is solved by assuming U— «PV+JQY, for we have then 
P(aPV+&QV)=AV-l-BHV, where A and B are given cubic functions of «, 5; and 
thence V=PU=AV+BHV, or A=l, B=0; the latter equation gives what is alone 
important, the ratio a:h', and it thus appears that there are three distinct cun'es of the 
third class U=0, and therefore (what indeed is shown in the Memoirs in LiouviUe) 
three distinct series of conics having the above-mentioned relation to the given curv e of 
the thu'd order V=0. 
It is hardly necessaiy to remark that the preceding theorems, although precisely 
analogous to those of Hesse, are entirely distinct theorems, that is the two series are not 
connected together by any relation of reciprocity. 
Article Nos. 24 to 28 . — Various investigations and theorems. 
24. Reverting to the theorem (No. 18), that the lineo-polar envelope of the line EP is 
the pair of lines OE, OF ; the line EF is any tangent of the Pippian, hence the theorem 
includes the following one : — 
The lineo-polar envelope with respect to the cubic, of any tangent of the Pippian, is 
a pau’ of lines. 
And conversely. 
The Pippian is the envelope of a line such that the lineo-polar envelope of the line 
with respect to the cubic is a pair of lines. 
It is I think worth while to give an independent proof. It has been shown that the 
equation of the lineo-polar envelope with respect to the cubic, of the line = 0 
(where t are arbitrary quantities), is 
_zv- 2 zr^, -vv-mn, w+vrx,, 2)^=0 ; 
and representing this equation by 
i(«, b, c,f, g, hjx, y, 2 )^= 0 , 
we find 
a5-A^=r(8ZT-f8ZV-r 
