406] 
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
223 
(3, i) 
(4) 
(•) 
= ?/ -f 2ft/ — 6 
(/) 
= 2ft 4- m — 6 
(•) 
= 1, 
(!) 
= i; 
which are the several cases for the conics which satisfy not more than four conditions, 
and 
69. 
For the 
(5) 
ft 
1) 
(3, 
2) 
(3, 
1, 
1) 
(2, 
3) 
(2, 
2, 
1) 
(2, 
1, 
1, 
(I, 
4) 
(Ï, 
1, 
3) 
A 
2, 
2) 
(Ï. 
1, 
1, 
(Ï, 
1, 
1, 
conics satisfying 5 conditions, we have 
= 1, 
= m 4- ft — 6, 
= — 9 + a, 
= f ft/ 2 4- 2 mn + |-w 2 — -tfz/z — 4- 27 — fa, 
= — 4 m — 4// — G 4- 3a, 
= 6m + 6/1 + 54 + a(m4-ft — 15), 
] ) = fm 3 4- m 2 n + run- 4- f ft 3 — f m 2 — 8mw — f n- 4- 2fm + — 75 
4- ot (- f m - f n + f£), 
= — 10m — 8/i — 5 4- 6a, 
= — 8m 2 — 12///M — 3/z 2 4- 60m 4- 57«. 4- 36 4- a (6?// 4- 3n — 45), 
= 27m 4- 24// 4- 27 — 23a 4- fa 2 , 
2) — - 4 f>-m 2 4- 30mn 4- f ] -n 2 — â f 1 m — — 189 
4- a (ft/ 2 4- 2m// 4- fft 2 — 27?// — 4- a f-) — f ot 2 , 
1, 1) = jL?/i 4 4- f ?// 3 // 4- m 2 // 2 4- f mzz 3 4- /j ft 4 — £ft/ 3 — 5m 2 n — 4m// 3 — f z/ 3 
— -i^-m 2 — 5mn — ^p// 2 4- -f-m 4- ft 4- 150 
4- a (— f m 2 — 3?//// — fzz 2 4- 4^m 4- -fn — 4- fa 2 . 
70. The given point on the curve to which the symbols 1, 2, &c. refer may be 
a singular 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/cl ; 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/cl ; 1/cl, 2/cl are thus special forms of I, 2, and the annexed T indicates 
the additional point of intersection arising ipso facto from the point 1 or 2 being a 
cusp. Similarly, we should have the symbols 3/cl, 4/cl, 5/cl; but it is to be observed 
that at a cusp of the curve there is no proper conic having a higher contact than