292 On the Surfaces of the Second Order. 



sines of the angles EFS and EFS (or of the angles EFSi and 

 EFS') are in a constant proportion to each other, because these 

 sines are proportional to those ratios. And since the right line 

 FE bisects the angles SFS' and S FSi, both internally or both 

 externally, in which case the angles SFS and S'FSi are equal, 

 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 SFS and S'FS (or of the halves of the 

 angles SFSi and S'FSi) is a consequent quantity. 



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

 passing through these points will approach indefinitely to a tan- 

 gent. Therefore when two surfaces are related as above, if a 

 right line passing through any point E of their common direc- 

 trix intersect one surface in the points S , Si, and touch the other 

 in the point S, the chord S Si will subtend a constant angle at 

 the common focus F, and this angle will be bisected, either in- 

 ternally 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 SFS or SFSi will be equal to the 

 ratio of the less value of m or ju 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 internal 

 or to the mean axis as directrix. In the latter relation 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 YW its mean axis perpendicular to the plane of 

 the focal lines. On one of the focal lines and its dirigent assume 

 any corresponding points F and A, and let AD be the directrix 

 passing through A. Then if a directive plane, drawn through 

 any point S of the surface, cut this directrix in D and the mean 



* See Exam. Papers, An. 1839, 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. 



