480 



portional to those ratios. And since the right line FE bi- 

 sects the angles SFS' and SqFSi, both internally or both 

 externally, in which case the angles SFSoand S'FSi areequal, 

 or else one internally and the other externally in which case 

 the angles SFS^ and S'FSi are supplemental, it is easy to 

 infer, from the constant ratio of the aforesaid sines, that in 

 the first case the product, in the second case the ratio of the 

 tangents of the halves of the angles SFSq and S'FSq (or of 

 the halves of the angles SFS] and S'FSi) is a constant quan- 

 tity. 



If the point S' approximate indefinitely to S, the right 

 line passing through these points will approach indefinitely 

 to a tangent. Therefore when two surfaces are related as 

 above, if a right line passing through any point E of their 

 common directrix intersect one surface in the points So, Si, 

 and touch the other in the point S, the chord SqSi will sub- 

 tend a constant angle at the common focus F, and this angle 

 will be bisected, either internally or externally, by the right 

 line FS drawn from the focus to the point of contact. And 

 the angle EFS being then a right angle, the cosine of the 

 angle SFSq or SFSi will be equal to the ratio of the less value 

 of m or |U to the greater.* 



§ 4. Among the surfaces of the second order the only one 

 which has a point upon itself for a modular focus is the cone, 

 the vertex of which is such a focus, related either to the in- 

 ternal or to the mean axis as directrix. In the latter rela- 

 tion the vertex belongs to the series of foci which are 

 ranged on the focal lines. To see the consequence of this, 

 let V be the vertex of the cone, and VW its mean axis 

 perpendicular to the plane of the focal lines. On one of 

 the focal lines and its dirigent assume any corresponding 



* See Exam. Papers, An. ]639, p. xxxi. questions 9, 10. These and some 

 of the preceding theorems were originally stated with reference to modular foci 

 only. They are now extended to umbilicar foci. 



