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XXXV. On certain New Theorems relative to the Conic Sec- 

 tions. By John William Stubbs, Esq.., M.A., formerly 

 of Trinity College, 'Dublin. 



IN the Number of the Philosophical Magazine published in 

 November 1843, the author proposed a new method of 

 deducing theorems relating to the right line and circle from 

 theorems generally known. The object of the present notice 

 is merely to show that all the theorems so deduced indicate 

 new properties of the conic sections. 



In order to make this manifest, the author borrows the 

 method used with so much success by M. Chasles in dedu- 

 cing the properties of cones of the second degree and of sphe- 

 rical conies, published in the sixth volume of the Transactions 

 of the Brussels Academy, in which he shows that all proper- 

 ties of the circle, which involve merely angles and positions, 

 lead by the consideration of the cyclic planes to properties of 

 cones having one cyclic plane in common, and of their reci- 

 procal cones with a common focal line, and hence to the sec- 

 tions of such cones by a sphere. 



The following theorems are believed by the author to be 

 new, as well as those relating to the circle from which they 

 are deduced; they will serve as examples of the fertility' of 

 these methods. 



I. If a point be assumed on the circumference of a circle 

 and an angle of constant magnitude turn round it, the chord 

 subtending this angle will envelope a circle. Now assuming 

 as a pole this point, and inverting the figure by producing the 

 radius until the rectangle under the original and whole pro- 

 duced radius is constant (Phil. Mag., Nov. 1843), we derive 

 the property, that if a right line and a point be given in posi- 

 tion, and if triangles have their vertical angle constant and 

 always at the given point, and their bases coinciding with the 

 given indefinite straight line, their circumscribing circles con- 

 stantly touch a fixed circle whose centre is in a straight line 

 perpendicular to the given one, and passing through the fixed 

 point. Now conceive the plane of the figure to be a common 

 section parallel to the cyclic plane of a number of cones having 

 one cyclic plane in common, and we deduce from the above 

 considerations, that if a plane (A) and right line (B) be given, 

 and if through the right line two planes be constantly drawn 

 to form with the given plane (A) a trihedral angle, and if the 

 traces of these planes on another given plane (C), passing 

 through the intersection of A and B, include a given angle, 

 the cone of the second order, of which C is a cyclic plane, 

 circumscribing this trihedral angle, will constantly touch a 

 fixed cone of the second order having the same vertex and 



