555] 
565 
555. 
NOTICES OF COMMUNICATIONS TO THE BRITISH ASSOCIATION 
FOR THE ADVANCEMENT OF SCIENCE. 
[From the Reports of the British Association for the Advancement of Science, I860 to 1873, 
Notices and Abstracts of Miscellaneous Communications to the Sections.] 
1. On the Problem of the in-ancl-circumscribed Triangle. Report, 1870, pp. 9, 10. 
I have recently accomplished the solution of this problem, which I spoke of at 
the Meeting in 1864. The problem is as follows : required the number of the triangles 
the angles of which are situate in a given curve or curves, and the sides of which 
touch a given curve or curves. There are in all 52 cases [see 514] of the problem, 
according as the curves which contain the angles and are touched by the sides are 
distinct curves, or are any or all of them the same curve. The first and easiest case is 
when the curves are all of them distinct; the number of triangles is here = 2aceBDF, 
where a, c, e are the orders of the curves containing the angles (or, say, of the 
angle-curves) respectively; and B, D, F are the classes of the curves touched by the 
sides (or, say, of the side-curves) respectively. An interesting case is when the angle- 
curves are one and the same curve; or, say, a = c = e (where the sign = is used to denote 
the identity of the curves); the number of triangles is here ={2a (a — l)(a — 2) + A) BDF, 
where a, A are the order and class of the curve a = c = e. In the reciprocal case, 
where the side-curves are one and the same curve, say B = D = F, we have of course 
a like formula, viz. the number of triangles is here = {2B (B — 1) (B— 2) + b] ace, where 
B, b are the class and order of the curve B = D = F. The last and most difficult case 
is when the six curves are all of them one and the same curve, say a=c=e—B=D=F; 
the number of triangles is here = one-sixth of 
A 4 ( . 
+ IX 
+ A 3 ( . 2a 3 - 18a 2 + 52a- 46) 
+ A 2 ( . - 18a 3 + 162a 2 - 420a + 221) 
+ A ( . 52q 3 - 420a 2 + 704a + 172) 
+ a 4 - 46a 3 + 221a 2 + 172a