288 



US to the opposite umbilic U'. Then, by reason of the sym- 

 metry of the surface, we shall have the angle SU'U equal 

 to SUiU, and supplemental to SUiU. Consequently the 

 tangents of the halves of SUUi and SU'U\ are to each 

 other in a constant ratio. From this it appears that the prin- 

 cipal ellipse passing through the umbilics is a common 

 asymptot to all the geodetic lines which pass through either 

 umbilic. 



" Though the umbilical geodetics are thus shown to be in- 

 finite spires, it is not true that all the geodetics on the surface 

 of an ellipsoid are of the same nature. Besides the geodetics 

 which coincide with the principal sections of the surface, there 

 are others among them which are closed curves. An exam- 

 ple will make this evident. 



" A circular disc, with a regular figure of an even num- 

 ber of sides inscribed in it, may be regarded as an infinitely 

 flattened spheroid of revolution, with a continuous geodetic 

 line traced upon its surface. In fact a closed cord, carried 

 along in the direction of the sides of the regular figure, and 

 passing over at each angle to the opposite side of the disc, 

 would be kept stretched round it. 



" I hope to be able, before long, to communicate to the 

 Academy a series of remarkable results respecting the com- 

 parison of similar geodetic arcs, at which I have arrived bv 

 means of the theorems stated in the beginning of the present 

 note. I expect also, by the translation of these geometrical 

 theorems into analytical language, to obtain some new rela- 

 tions between the integrals, to the consideration of which we 

 are led in the rectification of the geodetic lines and lines of 

 curvature of a surface of the second order." 



Rev. William Roberts communicated an analytic proof of 

 the theorem stated by Mr. Graves, and made some observa- 

 tions on different applications of the formula of M. Liouville, 

 on which the proof depends. 



