436 Professor Booth on the Focal Properties 



c b 

 (11.) The line OQ= , or the cord of (2) passing 



through a pair of conjugate foci = . 



(12.) The length of the perpendicular from one of the 

 foci of the surface on the corresponding directrix plane, is 

 c 3 

 a b £ * 

 (13.) The length of the perpendicular from the centre on 



one of the directrix planes = ^ — . 

 1 b e 



(H<.) The segment of the cord joining a pair of conjugate 



foci, intercepted between the plane of xy and one of the di- 



x. • i • c b 



rectnx planes, is = . 



a ri 



(15.) The length of the perpendicular from the focal centre 

 on one of the conjugate directrix planes == . 



We now proceed to give the enunciations of a very few 

 theorems, merely as specimens of the results which flow from 

 the preceding definitions, and the application of the method 

 alluded to above ; premising that neither the preceding de- 

 finitions, nor the following theorems, are applicable either 

 to the hyperboloid of one sheet, or to the hyperbolic parabo- 

 loid ; and this may suggest a natural division of surfaces of 

 the second order into two classes, the one containing the 

 umbilical surfaces, the other those surfaces whose generatrices 

 are right lines. 



Prop. I. — From any point t of a surface of the second 

 order, let perpendiculars^,^', be let fall on two conjugate di- 

 rectrix planes ; the rectangle under those perpendiculars is to 

 the square of r, — the distance of the point t from the focal 

 centre O, relative to those conjugate directrix planes in a con- 

 stant ratio, as the square of the perpendicular P from the 

 centre on one of the directrix planes is to the square of the 

 transverse axe a, or 



ppf P 2 



r 2 ~ « 2 ; 

 this constant ratio is one of equality, when the least semiaxis 

 c of the surface is equal to the perpendicular let fall from 

 the centre of the surface on the line joining the extremities 

 of the semiaxes a and b: this ratio of equality can never exist 

 then when the surface is one of revolution round the trans- 

 verse axes, except when (£) is an elliptic paraboloid. 



Generally when the ratio is one of equality, s 2 + if = 1. 



