434 Professor Booth on the Focal Properties 



through one of its umbilici, and let a point be assumed on 

 this diameter, at the distance u s from the centre ; this point 

 may be termed a focus of the surface. Hence in general a 

 surface of the second order has four foci, situated on the 

 umbilical diameters. 



(3.) Let a plane be drawn parallel to one of the circular 

 sections of the surface, meeting the umbilical diameter con- 



jugate to this circular section, at the distance — from the 



centre ; this plane may be termed a directrix plane of the 

 surface. 



This plane, and the focus of the surface, on the diameter 

 of the surface, conjugate to this plane, are polar plane and 

 pole, relative to the given surface. Hence in general a sur- 

 face (X) has four directrix planes, parallel two by two, to 

 its circular sections. 



Each pair of those planes intersect in the directrices of the 

 principal section, whose semiaxes are a and b; and these 

 pairs may be termed conjugate directrix planes. 



The foci of the surface, which are the poles of the conju- 

 gate directrix planes, may be called conjugate foci. 



The line joining a pair of conjugate foci is perpendicular to 

 the plane of the principal section in the plane of xy, whose 

 semiaxes are a and b. 



(4.) Let the middle point of this line be called the focal 

 centre of the surface. Hence in general a surface of the 

 second order has four foci and two focal centres. 



Thus C is the centre of (X). 



G and G' are the umbilici, s 

 the surface; A D, A D', the conjugate directrix pis 

 focal centre of (2) ; or the focus of the principal section of 



s and s' the conjugate foci of 

 nugate directrix planes ; O the 



