436 Professor Booth on the Focal Properties 



c b 

 (11.) The line O Q = - , 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 



a b s ' 

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



fl (* 



one of the directrix planes = -, . 



b e 



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

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



L. i c b 



rectnx planes, is = - . 

 a t\ 



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



be 

 on one of the conjugate directrix planes = -- . 



d 



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 tf, or 



ppf P* 



r 2 a* ; 



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 (5)) is an elliptic paraboloid. 



Generally when the ratio is one of equality, 3 -f- V 2 = 1 . 



