106 
PROFESSOR CAYLEY ON THE CURVES 
®0) 
=1, 
(•) 
=2 
(//) 
=1; 
(B,l) ( . } 
= w+2m— 6, 
(/) 
= 2w-f m— 6; 
(4) ( . } 
=1, 
(/) 
=1; 
which are the several cases for the conics which satisfy not more than four conditions, 
and 
69. For the conics satisfying 5 conditions, we have 
(5) =1, 
(4.1) 6, 
(3.2) = — 9+a, 
(3. 1. 1) =fm 2 +2mn+fw 2 — -^w-1-27— -fa, 
(2.3) = — 4m — in — 6 + 3a, 
(2.2.1) = 6m + 6w+54+a(w+w— 15), 
(2. 1. 1. 1) = + rtfn + » 2 + \n* — fm 2 — Smn — -§n 2 + — 75 
+a(— fm— f»+-^), 
(I, 4) = — 10m — 8n — 5 + 6a, 
(T, 1, 3) = — 8m 2 — 12mn— 3w 2 +60m+57w + 36-f- a(6m-f-3w— 45), 
(1,2,2) =27m+24rc+27-23a-ffa 2 , 
(1, 1, 1, 2) =- 2 %2 2 +30mw+ J ^ 2 — 189 
+ a(m 2 f- 2mw f- f w 2 — 2 7m — - 2 % + -^f^-) — f a 2 , 
(1, 1, 1, 1, l)=^ 2 -m 4 +fm 3 w+mV+fmw 3 +-^w 4 — fm 3 — 5m 2 n— Amrf— \v? 
-^m*-5mn-^^+^n+*iZn+150 
+ a( — f m 2 — 3m% — f% 2 + - 2 %i + — ^f^-) + f a 2 . 
70. The given point on the curve to which the symbols 1, 2, &c. refer may be a sin- 
gular point, and in particular it is proper to consider the case where the point is a cusp. 
I use in this case an appropriate notation ; a conic which simply passes through a cusp, 
in fact meets the curve at the cusp in two points ; and I denote the condition of passing 
through the cusp by 1*1 ; similarly, a conic which touches the curve at the cusp, in fact 
there meets it in three points, and I denote the condition by 2*1 ; 1*1, 2*1 are thus special 
