444 
and the principal plane of that cone passes through the direc- 
trix. 
‘© The above properties are true for every point on the wm- 
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 Mémoires, tom. 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 ealls 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 we, where wis the length of 
az 
— 2 
the umbilical semi-diameter, and «? = — (a>b>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, b) 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.” 
