WHICH SATISFY GIVEN CONDITIONS. 
109 
7 3. The remainder of this table, being the part where the symbols ( • ) and ( / ) do 
not occur, I present under a somewhat different form as follows : — 
=0, 
(4*1, 1) 
=0, 
(3x1, 2) 
= 0, 
(3^1, 1, 1) 
=0, 
(2,3) 
—(2x1, 3) 
=0, 
(2,2,1) 
— (2*1, 2, 1) 
~n — 3, 
(2, 1,1,1) 
—(2x1, 1,1, 1) 
=i(»-3)(»-4), 
(i, 4) 
-(1^1,4) 
=1, 
(1,1,3) 
— (1*1,1, 3) 
=(2*I, 3) + (»-3), 
(1,2,2) 
-(Ixi, 2,2) 
=3(w-3)+*-l, 
(1,1, 1,2) 
-(51,1,1, 2) 
=(2*1, 1, 2)+|{n-3)(7i-4)+J+2»-3 m- 
(1,1, 1,1,1) 
i-(lxl, 1,1, 1,1) 
=(2*I, 1, 1, 1). 
These results relating to a cusp, are useful for the investigations contained in the 
Second Memoir. 
It will be noticed that the symbols which contain 2*1 are not, like those which contain 
2, symmetrical in regard to (m, n) : the interchange of (m, n) would of course imply the 
change of a cusp into an inflexion, and would therefore give rise to a new symbol such 
as 2/1 ; but I have not thought it necessary to consider the formulae which contain this 
new symbol. 
Investigations in extension of those o/De Jonquieres in relation to the contacts of a 
Curve of the order r with a given curve. — Nos. 74 to 93. 
74. De Jonquieres has given a formula for the number of curves O' of the order r 
which have with a given curve U m of the mth order t contacts of the orders a, b, c, &c. 
respectively, which besides pass through p points distributed at pleasure on the curve 
IJ m (this includes the case of contacts of any orders at given points of the curve U m ), and 
which moreover satisfy any other — -—(a-\-b-\-c-\-&c.)—g> conditions; viz. the num- 
ber of the curves C r is =gj(a-\-l)(b +1)(<? + 1) . . . into 
[rm— (a-\-b-\-c . .)— jp ]' 
+[m— (a-\- b+c . .)— p— Yf~\a -\-b . OLD] 1 
' -\-\rm— (a-\-b-\-c . .)— p— 2] <_2 (a5+ac+Jc . .)[D] 2 
+[m— (a-\-b+c . .)— p — ff ( abc . . . 
r 2 
