379] 
549 
379. 
NOTICES OF COMMUNICATIONS TO THE BRITISH ASSOCIATION 
FOR THE ADVANCEMENT OF SCIENCE. 
[From the Reports of the British Association for the Advancement of Science, 1854 to I860, 
Notices and Abstracts of miscellaneous Communications to the Sections.] 
1. On the Solution of Cubic and Biquadratic Equations. Report, 1854, p. 1. 
2. On the Porism of the Tn-and-circumscribed Triangle. Report, 1855, p. 1. 
The porism of the in-and-circumscribed triangle in its most general form relates 
to a triangle the angles of which lie in fixed curves, and the sides of which touch 
fixed curves, but at present I consider only the case in which the angles lie in one 
and the same fixed curve which for greater simplicity I consider to be a conic. We 
have therefore a triangle ABC the angles of which lie in a fixed conic €> and the 
sides of which touch the fixed curves 21, 23, (£. And if we consider the conic <5 and 
the curves 21, 25 as given, the curve (£ will be the envelope of the side AB of the 
triangle. Suppose that the curves 21, 23 are of the classes to, n respectively; there is 
no difficulty in showing that the curve (£ is of the class 2mn. But the curve (5 has 
in general double tangents forming two distinct groups, the first group arising from 
the quadrilaterals inscribed in the conic @ and such that two opposite sides touch 
the curve 21, and the other two opposite sides the curve 23; the second group arising 
from quadrilaterals such that two adjacent sides touch the curve 21 and the other 
two adjacent sides touch the curve 23. The number of double tangents of the first 
group is = mn (mn — 1), and the number of double tangents of the second group is 
= mn (mn — to — n + 1) ; the number of double tangents of the two groups is therefore 
= mn (2mn — to — n). The curve (S has not in general any inflexions, hence, being of 
the class 2mn with mn (2mn — m — n) double tangents, it will be of the order 
2mn (m + n — 1).