43 



so that the condition (X") may be thus written : 

 or, by a further transformation, 



^ being here the distance of the point x y z upon the ray, beyond the 

 centre of the spheroid. This equation (Z") contains the law of oscu- 

 lation of the spheroid, since it expresses the dependence of the dis- 

 tance § on the direction of the plane passing through the ray and 

 through the consecutive point x -{• ix, y + ^i/, z + h. We shall 

 call this plane the plane of osculation of the spheroid ; and we see, 

 by comparing (Z") with (C), that the extreme values of § corres- 

 pond to those directions of the plane of osculation in which it touches 

 the developable pencils ; while the corresponding extreme positions of 

 the centre of the osculating spheroid, are contained upon the caustic 

 surfaces. And when the ray is one of those principal rays deter- 

 mined in the preceding number, it is easy to prove that the equation 

 (Z") is satisfied independently of the ratios of the differentials, if we 

 assign to § the value which belongs to the principal focus ; the prin- 

 pal foci are therefore the centimes of spheroids, which have complete 

 contact of the second order with the surfaces of constant action. 

 The equations which express this property of the principal foci are 



G 2 



