226 
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
[406 
(2*1, 1, 1) 
( • ) = ?n 2 + 2m?i + fyi 2 — 7m — tyn -1- 21 — fa, 
( I ) = ^m 2 + 2mw + w 2 —-fm — 7w + 18 — |a. 
73. The remainder of this table, being the part where the symbols (•) and (/) 
do not occur, I present under a somewhat different form as follows: 
(5*1) 
= 0, 
(ia, i) 
= 0, 
(3*1, 2) 
= 0, 
(3*1, 1, 1) 
= 0, 
(2, 3) 
-(2*1, 3) 
= 0, 
(2, 2, 1) 
- (2*1, 2, 1) 
= n — 3, 
(2, 1, 1, 1) 
-(2*1, 1, 1, 1) 
= hi n - 3) ( n - 4), 
Æ 4) 
-(1*1, 4) 
= 1, 
(Ï, 1, 3) 
- (1*1, 1, 3) 
= (2*1, 3) + (n — 3), 
(Ï. 2, 2) 
-(1*1, 2, 2) 
= 3 (n — 3) + * — 1, 
(ï, T 1, 2) 
-(1*1, 1, 1, 2) 
= (2*1, 1, 2) + *(n- 
(I, 1, 1, 1, 
1) - (1*1, 1, 1, 1, 
1) = (2*1, 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 2tl; but I have not thought it necessary to consider the formulae 
which contain this new symbol. 
Investigations in extension of those of De JoNQUiiSRES in relation to the contacts of a 
Curve of the order r with a given curve. Article Nos. 74 to 93. 
74. De Jonquieres has given a formula for the number of curves C r of the 
order r which have with a given curve XJ 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 U m (this includes the case of contacts of any orders at given points of 
the curve U m ), and which moreover satisfy any other \r (r + 3) — (a 4- b + c + &c.) — p