384 
ME. A. CAYLEY ON THE CONIC OE EIYE-POINTIC 
Proof of corresponding expressions for (m— 1)®Q|, (m— 1)"Q4:- 
27. We have 
= — 9(m — 2)'H^O 
4-3(m-2)Hn^ 
= — 6(w— 2)(??i~ 3)HO 
+2(?7i-3)n^ 
To which I join 
(m— 1/Q2=H. 
28. We have consequently 
+ 3(TO-4)Hn& 
+ (~3aH+T)^h 
and 
(to— 1)®(3(to— 2 ) 0^04 ■~2(to— 3)Q1) = (— 3(to— 2)nH+2(«i— 3)T)^h 
which for to=: 4 become 
729(3QA~QI)=(-3nH+'T)^^ 
29. In the case to =4, we have Hesse’s theorem, that the equation 3 Q.,Q 4 — Qs=0 
gives a curve of the 14th order, which passes through the points of contact of the double 
tangents, viz. substituting for O, T their values, the equation of this cul^e is 
-3H(9[3, C, 
+ (a, C, Jf, #, b,H)^=0. 
I have added these remarks for the sake of pointing out the striking resemblance of the 
expressions which occur in the double-tangent problem for ??i=4, and in the piesent 
theory of the five-pointic conic for any value whatever of to. It has not hitherto been 
shown what the expressions 3 ( to — 2)Q2Q4 — 2(w— 3)Q,j and — o(to— 2)nH-4-2(TO— o)T 
respectively denote, except in the particular case ni 4. 
III. Application of the Formula.’ to the Cubic. 
30. I shall apply the formula for the five-pointic conic to a cubic ; to avoid confusion 
to a numerical factor, I write XT', H' in the place of U, H, so that we have 
1 [-3(913, IP. P. ,djHr.H'| 
+ 35, C. Jf, (g, Vn J’ 
