On the Surfaces of the Second Order. 289 



consequently bear a given ratio to SF, the distance of the point 

 S from the focus. And the same thing will be true when the 

 directrix and focus are umbilicar, because the perpendicular dis- 

 tance of the point S from the directrix will be in a constant ratio 

 to its distance from each directive plane drawn through the 

 directrix. 



The fixed plane of section will in general contain another 

 directrix parallel to the former, and belonging to the same 

 focal ; and it is evident that the perpendicular distance of S from 

 this other directrix will be in a given ratio to its distance SF' 

 from the corresponding focus F 7 , the ratio being the same as in 

 the former case. Hence, according as the point S lies between 

 the two directrices, or at the same side of both, the sum or dif- 

 ference of the distances SF and SF' will be constant. 



If the plane of section pass through either of the foci, as F, 

 this focus and its directrix will manifestly be the focus and di- 

 rectrix of the section. In this case the plane of section will be 

 perpendicular to the focal at F. And if the surface be a cone, 

 the point F being anywhere on one of its focal lines, the distance 

 of the point S from the directrix will be in a constant ratio to 

 its perpendicular distance from the dirigent plane which con- 

 tains the directrix, and therefore this perpendicular distance will 

 be in a given ratio to the distance SF. Now, calling Y the 

 vertex of the cone, and taking SY for radius, the perpendicular 

 distance aforesaid is the sine of the angle which the side SY of 

 the cone makes with the dirigent plane ; and SF, which is per- 

 pendicular to YF, is the sine of the angle SYF. Consequently 

 the sines of the angles which any side of a cone makes with a 

 dirigent plane and the corresponding focal line are in a given 

 ratio to each other. 



2. Conceive a surface of the second order to be intersected 

 in two points S, S' by a right line which cuts two parallel direc- 

 trices in the points E, E', and let F, F 7 be the foci correspond- 

 ing respectively to these directrices. The perpendicular dis- 

 tances of the points S, S' from the first directrix and from the 

 second are to each other as the lengths SE, S'E, SJE', S'E' 



u 



