227 
406] ON THE CURVES WHICH SATISFY GIVEN CONDITIONS, 
conditions; viz. the number of the curves G r is = p (a + 1) (b + 1) (c + 1)... into 
[rm - (a + b + c..)-p Y 
+ [rm-(a + b + c..)-p~l] t ~ 1 (a + b +c ..) [D] 1 
+ [rm — (a + b + c..) — p — 2] i_2 (ab + ac + be ..) [D] 2 
, -t- [rm - (a+ b + c .-p - t]° (abc ... ) [D] f , 
where the curve \J m is a curve without cusps, and having therefore a deficiency 
D = \ (m — 1) (m — 2) — S ; 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 = b, &c., then the foregoing 
expression is to be divided by [a] 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 a + 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 hr (r + 3) — (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 formula; of the present section. 
75. 1 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 p = 1. 
. We have thus a curve G r having with the given curve U m t contacts of the 
orders a, b, c.. respectively, and besides passing through \r (r + 3) — (a + b + c..) arbitrary 
points; and the number of such curves is by the formula = (a + 1) (b + 1)(c + 1),... into 
( [rm — (cH- b + c..) Y 
J + [rm — {a + b + c..) — l] i_1 (a +b + c ..) [D] 1 
j + [rm — (a + b + c..) — 2] i-2 (ab + ac +be ..) [D] 2 
[ + [rm — (a + b + c.— tj (abc... ) [D] i , 
where, as before, in the case of any equalities between the numbers a, b, c,..., the 
expression is to be divided by [a]“ [/3] 3 .... 
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, &c.; 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 + 1) (rm - a \ 
t+aZ) J 
— a . . k: 
29—2