396 
ME. A. CAYLEY 0]S' THE COYIC OE ERTE-POIYTIC 
Hence, multiplying by H and substituting this value of TH, the equation becomes 
SHn^V - 6HmV + 4WHi - 4Y||i = 0, 
ov, as we may write it, = 0 ; 
and we can at once make the equation divide by V, viz. by assummg 
n 
__2Mi 
-LiAV. 
where A is arbitrary. We have thus a four-pointic contact. And substituting for H. 
and throwing out the factor V, the equation becomes 
3H(^|^+iAV) — 4Bi+2W(fHA) = 0; 
01 leducing, j_36H*AW+24HH,AV+9H*AA ’=0, 
which is the equation'of the lines drawn from the point of contact to the remaining two 
points of intersection. 
52. I write for greater convenience 
„ @ . 
0 being as yet indeterminate, the equation is thus reduced to 
And we have then to determine 0 so that the left-hand side may divide by V; or, what 
is the same thing, we must determine 0 so that 
H(H?-3H1.) + 0AV 
may divide by V. This implies the existence of an identical equation, 
H(Hf-3Hl,)+0W=MU+NV, 
which for U=0 would give the decomposition in question; but I have not investigated 
the values of M and N. I assume at the outset U = 0, and putting, as be oie, 
and writing also 
S = Xa'[( 1 + 8 -f ( 4Z + 41 )xhjz + ( — 2 + 2 Vy/z' ] 
_l_ Y?/[(l + U^)if + (4^+ ( - 2Z^ 
-\-Zz [(1 + 8Z'K +(4Z+41Z"K^i/+(-2ZH2Z^)Ty], 
I remark that for U=0 we have 
+ VS + (l-b 8Z^)(QW - = 0, 
an equation which, observing that 
and assuming also 
H=— (l + SZ'^Y^, 
0 = (l + 8ZyQ, 
