444 



and the principal plane of that cone passes through the direc- 

 trix. 



" The above properties are true for every point on the um- 

 bilicar focal of the surface, with the directive planes and direc- 

 trix corresponding to that point. When the two directive planes 

 coincide, these theorems, suitably modified, reduce to known 

 properties of the non-modular surfaces of revolution. For that 

 particular case they have been demonstrated by M. Chasles, 

 in the Transactions of the Royal Academy of Brussels {Nou- 

 veaux Memoires, torn, v.) In the general shape in which 

 they are, I believe, now for the first time* given, they appear 

 to me of sufficient elegance to merit the attention of geome- 

 ters." 



* " Since the above note was read, my attention has been directed to a paper 

 ' On the Focal Properties of Surfaces of the Second Order,' by Dr. Booth, 

 in the Philosophical Magazine for December, 1840. In that paper he consi- 

 ders as analogous to the foci in conic sections four points, which he calls the. 

 foci of the surface, situated, two by two, on the umbilical diameters, at dis- 

 tances from the centre equal to each other and to ut, where u is the length of 



a2 c2 



the umbilical semi-diameter, and t* =: — — - (a > 6 > c.) The polar planes 



of these points he terms the ' directrix planes' of the surface, and of these 

 planes the two which intersect in a directrix of the principal section (a, h) are 

 ' conjugate directrix planes.' The foci of the same section (a, b) he calls 

 the ' focal centres' of the surface. These definitions being premised, he 

 states the theorem, that if from any point of the surface perpendiculars be 

 let fall on two conjugate directrix planes, the rectangle under those perpen- 

 diculars is to the square of the distance of the point from the corresponding 

 focal centre in a constant ratio. But he does not observe the fact which gives 

 the umbilicar generation its chief interest and value, namely, that the ♦ focal 

 centre' may traverse the focal curve on which it lies, the ' directrix planes' 

 changing along with it, while the generated surface remains unaltered. He 

 then proceeds to state several properties of his ' focal centres' and ' direc- 

 trix planes,' and among them I find those which I have marked (1), (2), (8), 

 and (9). But these theorems are given by him only for his two ' focal cen- 

 tres' and his four ' directrix planes ;' whereas they are really properties of 

 every point on the umbilicar focal of the surface, and the directive planes 

 corresponding to such point." 



