485] 
PROBLEMS AND SOLUTIONS. 
547 
69—2 
conditions which relate to intersection at an unascertained point (of course the inter 
sections referred to must be at least 2-pointic, for otherwise there is no condition at 
all) we may consider the conics which pass through four points and satisfy one con 
dition ; or which pass through three points and satisfy two conditions; or which pass 
through two points and satisfy three conditions; or which pass through one point and 
satisfy four conditions; or which satisfy five conditions. Considering in particular the 
last case, let 1 denote that the conic has 2-pointic intersection, 2 that it has 3-pointic 
intersection, ... 5 that it has 6-pointic intersection with a given curve at an unascertained 
point. 
Then the problems are in the first instance 
5 ; 4, 1; 3, 2 ; 3, 1, 1 ; 2, 2, 1 ; 2, 1, 1, 1; 1, 1, 1, 1, 1. 
But the intersections may be intersections with the same given curve or with different 
given curves; and we have thus in all 27 problems, viz. these are as given in the 
following table, where the colons (:) separate those conditions which refer to different 
curves: 
No. of 
Prob. 
Conditions. 
No. of 
Prob. 
Conditions. 
No. of 
Prob. 
Conditions. 
1 
5 
10 
3, 
1 
1 
19 
3 
1 
1 
2 
4, 
1 
11 
3 
1, 
1 
20 
2 
2 
1 
3 
3, 
2 
12 
9 
u 9 
2 
1 
21 
9 
1 : 
1 
1 
4 
3, 
1, 
1 
13 
9 
1 
2 
22 
2 
1, 
1 
1 
5 
2, 
9 
1 
14 
2, 
1, 
1 
1 
23 
1, 
1, 
1 
1 : 
1 
6 
9 
b 
1, 
1 
15 
2, 
1 : 
1, 
1 
24 
1, 
1 
1, 
1 : 
1 
7 
1, 
1, 
1, 
1, 1 
16 
2 
1, 
b 
1 
25 
2 
1 
1 
1 
8 
4 
1 
17 
1, 
1, 
1, 
1 : 1 
26 
b 
1 : 
1 
1 : 
1 
9 
3 
2 
18 
1, 
1, 
1 
1, 1 
27 
1 
1 : 
1 
1 : 
1 
Thus Problem 1 is to find a conic having 6-pointic intersection with a given curve : 
Problem 2 a conic having 5-pointic intersection and also 2-pointic intersection with a 
given curve... Problem 7 is to find a conic having five 2-pointic intersections with 
(touching at five distinct points) a given curve....Problem 27 is to find a conic having 
2-pointic intersection with (touching) each of five given curves. Or we may in each 
case take the problem to be merely to find the number of the conics which satisfy 
the required conditions. This number is known in Prob. 1, for the case of a curve 
of the order m without singularities, viz. the number is = m(12m —27). It is also 
known in Problems 25 and 26 in the case where the first curve (that to which the 
symbol 2, or 1, 1 relates) is a curve without singularities ; and it is known in Prob. 27, 
viz. if m, n, p, q, r be the orders and M, N, P, Q, R the classes of the five curves