110 
PEOFESSOE CAYLEY ON THE CTJEVES 
where the curve U m is a curve without cusps , and having therefore a deficiency 
V)(m— 2) — h; the numbers a, b, c, . . are assumed to be all of them unequal, 
but if we have a of them each = a, (3 of them each =5, &c., then the foregoing expression 
is to be divided by [«] a [/3] ? . . . ; and p denotes the number of the curves C r which satisfy 
the system of conditions obtained from the given system by replacing the conditions of 
the t contacts of the orders a, b , c, &c. respectively by the condition of passing through 
ci-\-b-\-c . . . arbitrary points. In order that the formula may give the number of the 
proper curves C r which satisfy the prescribed conditions, it is sufficient that the 
\r{r-\-S) — (a-\-b-\-c . .)—p conditions shall include the conditions of passing through at 
least a certain number T of arbitrary points : this restriction applies to all the formulae 
of the present section. 
75. I will for convenience consider this formula under a somewhat less general form, 
viz. I will put p=0, and moreover assume that the %r(r-\-3) — {a-\-b-\-c . .) conditions 
are the conditions of passing through this number of arbitrary points; whence ^=1. 
We have thus a curve C’’ having with the given curve U m t contacts of the orders 
a, b, c . . respectively, and besides passing through ^r(r-\-S) — {a-\-b-\-c . .) arbitrary 
points ; and the number of such curves is by the formula=(a+l)(5+l)(c+l), . . . into 
[m— (a+J+c . .) y 
+[rm— {a-\-b-\-c . .)— l] i-1 (« -\-b -\-c . .)[D]‘ 
' -\-[rm— (a-\-b+c . .)— 2 y~\ab-\-ac-\-bc . -)[D] 2 
[ +[m — . .)— #]° ( abc . . . )[D]*, 
where, as before, in the case of any equalities between the numbers a, b, c , . . . , the expres- 
sion is to be divided by [a]“[i3] p . . . 
76. I have succeeded in extending the formula to the case of a curve with cusps: 
instead of writing down the general formula, I will take successively the cases of a single 
contact a, two contacts a, b , three contacts a, b, c, See. ; and then denoting the numbers 
of the curves C r by (a), (a, b), (a, b, c ), &c. in these cases respectively, I say that we 
have 
(a)= {a-\-l)^rm— 
l+«D 1 
—a . . 
(a, b)= (a-\-l)(b-\-l) f [rm—a—by 
\ +[rm— a— b— 1]‘(«+ £)[D]‘ 
1+ ah [DJ 
— ’ «(£+l)[[m— a— b— l] 1 1 
1+ VD\ 
-\-b(a-\-l) J[rm— a— l] 1 ] 
1+ «DJ 
-{- (ib ........... 
