467 



of the distance of the same point from the focus ; and, as the 

 locus of a point, whose distance from a given plane is pro- 

 portional 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 distance of any point of 



* In attempting to find a geometrical generation for the surfaces of the se- 

 cond 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 dis- 

 tance from a given point should be 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 inde- 

 pendently, 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 pro- 

 perty 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 Sur- 

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

 of Paris, on the 26th December, 1642, 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 im- 

 portance 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 l' Academie 

 des Sciences, tom. xvi. pp. 783-828, 885-890; April, 1843); and also occa- 

 sioned several discussions, principally between M. Poncelet and M, Chasles, 

 relative to that Memoir (^Comptes rendus, tom. xvi. pp. 829, 938, 947, 110.5, 

 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 pro- 

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



