280 On the Surfaces of the Second Order. 



of the distance of F from any point of this surface is in a con- 

 stant ratio to the rectangle under the distances of the latter 

 point from the two planes. And these planes are directive 

 planes ; because, if a section parallel to one of them be made 

 in the surface, the distance of any point of the section from the 

 other plane will be proportional to the square of the distance of 

 the same point from the focus ; and, as the locus of a point, 

 whose distance from a given plane is proportional to the square 

 of its distance from a given point, is obviously a sphere, it 

 follows that the section aforesaid is the section of a sphere, and 

 consequently a circle ; which shows that the plane to which the 

 section is parallel is a directive plane. Thus,* the square of the 



r * In attempting to find a geometrical generation for the surfaces of the second 

 order, one of the first things which I thought of, before I fell upon the modular 

 method, was to try the locus of a point such that the square of its distance from a 

 given point should he in a constant ratio to the rectangle under its distances from 

 two given planes; but when I saw that this locus would not represent all the 

 species of surfaces, I laid aside the discussion of it. Some time since, however, 

 Mr. Salmon, Fellow of Trinity College, was led independently, in studying the 

 modular method, to consider the same locus ; and he remarked to me, what I had 

 not previously observed, that it offers a property supplementary, in a certain sense, 

 to the modular property; that when the surface is an ellipsoid, for example, the 

 given point or focus is on the focal hyperbola, which the modular property leaves 

 empty. This remark of Mr. Salmon served to complete the theory of the focals, 

 by indicating a simple geometrical relation between a non-modular focal and any 

 point on the surface to which it belongs. 



In a memoir " On a new method of Generation and Discussion of .the Surfaces 

 of the second Order," presented by M. Amyot to the Academy of Sciences of Paris, 

 on the 26th December, 1842, the author investigates this same locus, conceiving it 

 to involve that property in surfaces which is analogous to the property of the focus 

 and directrix in the conic sections ; and the importance attached to the discovery of 

 such analogous properties induced M. Cauchy to write a very detailed report on 

 M. Amyot' s memoir, accompanied with notes and additions of his own (Comptes 

 rendus des Seances de V Academic des Sciences, torn. xvi. pp. 783-828, 885-890 ; 

 April, 1843) ; and also occasioned several discussions, principally between M. 

 Poncelet and M. Chasles, relative to that Memoir (Comptes rendus, torn. xvi. 

 pp. 829, 938, 947, 1105, 1110). But the property involved in this locus cannot 

 be said to afford a method of generation of the surfaces of the second order, since 

 it applies only to some of the surfaces, and gives an ambiguous result even where 

 it does apply. It is therefore not at all analogous to the aforesaid general property 

 of the conic sections, and moreover it was not new when M. Amyot brought it 



