379] 
BRITISH ASSOCIATION FOR THE ADVANCEMENT OF SCIENCE. 
553 
9. On the Problem of the In-and-circumseribed Triangle. Report, 1864, p. 1. 
The general problem of the in-and-circnmscribed triangle may be thus stated, viz., 
to find a triangle the angles whereof severally lie in, and the sides severally touch, a 
given curve or curves; and we may, in the first instance, inquire as to the number 
of such triangles. The first and easiest case is when the curves are all distinct; here, 
if the angles lie in curves of the orders m, n, p, respectively, and the sides touch 
curves of the classes Q, R, S, respectively, then the number of triangles is = ZmnpQRS. 
The number may be obtained for some other cases; but the author has not yet 
considered the final and most difficult case, viz. that in which the angles severally 
lie in, and the sides severally touch, one and the same given curve. 
The foregoing notices relate to verbal communications upon questions with which 
I was at the time occupied and which are for the most part more fully discussed in 
papers printed elsewhere. I remark upon them as follows: 
1. I have no remembrance as to this; I think no paper printed or written. 
2. See 175. 
3. See 158. 
4. No paper printed. The intention was to consider the different forms of 
trinodal quartic curves, in particular those with real nodes, as obtained from the 
inversion of a conic according to the different relations of the conic to the fundamental 
triangle. Thus according as the conic cuts in two real points, touches, or cuts in two 
imaginary points, a side of the triangle, the tangents at the corresponding node are 
real, coincident, or imaginary; viz. the node is a crunode, cusp, or acnode. And in 
the case of real intersections there is a further distinction according as the inter 
sections lie each or either of them on the side itself, or on the side produced in 
one or other of the two directions. By considering the different relations of the conic 
to the fundamental triangle we thus obtain the different forms of the trinodal quartic. 
5. See 351. 
6. I think no paper printed or written. 
7. See 302 and 305. 
8. See 306. 
9. The question is considered in a memoir On the Problem of the In-and-circum- 
scribed Triangle, Phil. Trans, t. clxi. (for 1871), pp. 369—412. 
C. V. 
70