284 



of the surface on the tangent plane passing through one of those 

 sides, and D the semidiameier parallel to that side. 



" If F, the vertex of the cone, be supposed to approach in- 

 definitely near to a point on the surface, its axes ultimately 

 coincide with the normal at that point, and the tangents to 

 the lines of greatest and least curvature passing through it. 

 The plane of two successive elements of a geodetic line 

 through V contains the normal, and those elements are equally 

 inclined to it. So likewise the plane of two successive ele- 

 ments of a line of curvature passing through /^contains the 

 tangent to that line of curvature at V; and the elements them- 

 selves are equally inclined to the normal. In virtue of the pre- 

 ceding proposition we are, therefore, entitled to conclude, that 

 the quantity PD remains unaltered for two successive ele- 

 ments, either of a geodetic line or of a line of curvature traced 

 on a central surface of the second order. 



" Returning to the case in which the vertex of the cone 

 is at a finite distance from the surface, we may now say that 



''Ifa central surface of the second order (A) be circum- 

 scribed by a cone (a), the quantity PD is the same for the 

 geodetic lines which are the prolongations of \->, L', L", L'", 

 four sides of the cone which make equal angles with its inter- 

 nal axis. 



" In what follows I shall suppose the surface {A) to be 

 an ellipsoid, in order to avoid the enumeration of a variety of 

 cases. The conclusions arrived at may, however, be adapted 

 to the hyperboloids by obvious modifications. 



" The four sides of the cone, denoted according to their 

 order by L, L', L", L'", being all tangents to geodetics on 

 the ellipsoid for which PD is the same,* are likewise all tan- 

 gents to the same confocal hyperboloid (i?), which intersects 

 the surface (^A) in the pair of opposite lines of curvature 



* I think it would be convenient, in future, to designate geodetic lines 

 of this kind as similar geodetics. 



