TRANSACTIONS OF SECTION A. 



571 



TIence mode of dcscribincr confocal ellipses by placing stricgs round an ellipse 

 and keeping them stretched by a pencil point. 



And, similarly if an endless string be placed round an ellipse whose circum- 

 ference is shorter than the string, and if the point 51' be fixed, and a loop of the 

 string be passed through a small ring at L and pulled tight, then if a pencil be put 

 through the ring and moved steadily away from the ellipse, a confocal hyperbola 

 will be described (fig. 5). 



Various other cases arise. 



7. As long as T is on a confocal ellipse PT— PL is constant. 



Also the tangents from T to the circle inscribed in TQQ/ are of constant length. 



If the tangent at L meet the confocal ellipse through T in T' and T", the 

 confocal hyperbblas through T','T" will pass through P, P' respectively and T' T" 

 — arc PP' is constant. 



Eight equal tangents can be drawn from the outer to the inner of two confocal 

 ellipses. 



8. If in fig. 5 the tangent be drawn at M, M' or L' to meet the tangent from 

 T in Q and Q', then the point of contact of this tangent will be the point at 

 which either an inscribed or an escribed circle of the triangle TQQ' will touch the 

 ellipse. 



9. If a triangle be drawn with each side touching an ellipse, then an infinite 

 number of triangles of equal perimeter can be drawn whose sides touch the ellipse ; 

 their three angular points lying one on each of three confocal ellipses. 



10. If the "three* confocals which are the loci of the angular points coincide, the 

 triangles are inscribed in one and circumscribed about another of two confocal 

 ellipses. 



In this case their perimeter is a maximum for all triangles inscribed in the 

 outer ellipse, and a minimum for all triangles circumscribing the inner ellipse. 



11. So for a polygon of any number of sides. 



Also each tangent is divided into two sections at the point of contact. If the 

 polygon has n sides there are 2?i. sections; starting with anyone of them and 



Fig. 5. 



Fig. 6. 



numbering the sections in order from 1 to n, the remaining sections are equal in 

 magnitude to those already passed, occurring in the same order. The sections of the 

 sides of the triangle in fig. 6 with the same number attached are equal to one another. 



10. Some Tangential Transformations, including Laguerre's Semi-Droites 

 Eeciproques. By Professor R. W. Genese, M.A. 



The equation to a straight line rutting off a length a from the positive axis of x 

 and making an angle cot - 'wj with that axis, being .r - a = ?«y, a relation of the 

 form 



ani^ -^ 2/ima + bu- + 2(/7n + 2fa + c = . . • (1) 



