274 



several authorities on these facts, and some curious allusions 

 to them in ancient Irish manuscripts.* 



Dr. A. S. Hart read a paper on the form of geodesic lines 

 through the umbilic of an ellipsoid. 



If u) be the angle at the umbilic of an ellipsoid, between 

 the principal section of the surface and any other geodesic 

 line, and if 9 be the angle between the plane of the principal 

 section through the umbilics and the osculating plane of this 

 geodesic line, at any point A, and if a be the semi-angle of 

 the right cone circumscribing the ellipsoid at the point A, a, 

 b, and c being the semi-axes of the ellipsoid, the angle 9 may 

 be determined by the following equation : 



tan i 9 v'taJ- 62)4*8-0) 



i: 



_ y(a2tani!a+6*)/(c«tan8a+ft8;- 



tan ^ a» 



Hence it follows that, as this line passes and repasses for ever 

 through the two opposite umbilics, the tangents of the halves 

 of the angles which it makes at these points with the plane of 

 the umbilics will be a series of continued proportionals, the 

 coefficient of the common ratio being determined by making 



a = - -g in the above equation. 



If c = 0, the ellipsoid becomes a plane ellipse, and the geo- 

 desic line becomes the focal radius vector ; and, the curvature 

 being infinite at the circumference, it passes through the 

 other focus, and so on for ever, forming, as before, a series of 

 angles, such that the tangents of their halves are a series of 

 proportionals. 



* Dr. Petrie's communication will appear in full in a subsequent number 

 of the Proceedings. 



