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MATHEMATICS AND PHYSICS. 



Mathematics. 



On the Porism of the in-and-circumscribed Triangle. 

 By A. Cayley, M.A., F.R.S. 



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 assume 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 ^,38, C And if we consider the conic §> and the 

 curves %, 33 as given, the curve C will be the envelope of the side AB of the triangle. 

 Suppose that the curves ^, 33 are of the classes m, n respectively, there is no diffi- 

 culty in showing that the curve C is of the class 2mn. But the curve C has in 

 general double tangents, forming two distinct groups, the first group arising from 

 quadrilaterals inscribed in the conic ^, and such that two opposite sides touch the 

 curve ^ and the other two opposite sides touch the curve 33, the second group 

 arising from quadrilaterals inscribed in the conic ^, and such that two adjacent sides 

 touch the curve 'M and the other two adjacent sides touch the curve 33. The number 

 of double tangents of the first group is mn (rnn — 1), and the number of double tan- 

 gents of the second group is mn (ran — m — w-j-1) ; the number of double tangents of 

 the two groups is therefore m n (2mn — m — n). The curve C has not in general any 

 inflexions, hence being of the class 2mn and having mn (Jlmn — m — n) double tan- 

 gents, it will be of the order 2mn (m-j-w — 1). 



When the curves ^ and 33 are conies, the curve C is therefore of the class 8, with 

 16 double tangents but no inflexions, consequently of the order 24. But there are 

 two remarkable cases in which the order is further diminished. First, when each of 

 the conies ^, 33 has double contact with the conic ^. The four points of contact 

 give rise to 8 new double tangents, or there are in all 24 double tangents, the curve 

 C is therefore of the degree 8 ; and being of the class 8, with 24 double tangents, it 

 must of necessity break up into four curves each of the class 2, i. e. into four conies. 

 Each of these has double contact with the conic §}, or attending only to one of the 

 four conies, we have the well-known theorem, which 1 call the porism (homographic) 

 of the in-and-circurascribed triangle, viz. " there are an infinity of triangles inscribed 

 in a conic, and such that the sides touch conies having each of them double contact 

 with the circumscribed conic." 



Secondly, the conies ^ and 33 may intersect the conic ^ in the same four points. 

 Here every tangent of the curve C is in fact a double tangent belonging to the first 

 mentioned group, the curve C in fact consists of two coincident curves ; each of them 

 therefore of the class 4. But this curve of the class 4 has itself four double tangents, 

 arising from the common points of intersection of the conies 3, 33 with the conic ^ ; 

 it must therefore break up into two curves, each of the class 2, i. e. into two conies ; 

 each of these intersects the conic ^ in the same four points in which it is intersected 

 by the conies ^, 38. Attending only to one of the two conies, we have the other well- 



1855. 1 



