287 



of the coixl ; whilst the parts which coincide with geodetic 

 lines on {A) roll, one of them on, and the other off, its cor- 

 responding line of curvature. 



" If, on the other hand, we consider a continuous cordp Vp 

 coincident with a pair of adjacent sides of the cone, as L and 

 L', we see that the locus of the style at V, which keeps it 

 stretched, will be a line of curvature on a confocal ellipsoid, if 

 the conditions of motion be the same as before. 



"The theorems here announced are meant to take the place 

 of two which were incorrectly given at page 192. I fell into 

 an error in the statement of them, partly by my haste in ge- 

 neralizing from particular cases in which they are true; and 

 partly in consequence of my having formed an inaccurate 

 conception of the form of geodetic lines in general. 



" For clearer views as regards this latter point, I acknow- 

 ledge myself indebted to the recently published researches of 

 my friend, Dr. Hart. 



" I may be allowed to take the present opportunity of men- 

 tioning, that a theorem lately announced by him to the Aca- 

 demy, respecting the form of a geodetic line which passes 

 through an umbilic, may be derived geometrically from a 

 theorem discovered by Mr. Michael Roberts. 



" Mr. Roberts has shown that if two geodetic lines be drawn 

 from the interior umbilics of a line of curvature of an ellip- 

 soid to the same point on the curve, the product of the tan- 

 gents of the halves of the angles which they make with the 

 arc of the principal ellipse joining the umbilics is constant. 



" Suppose now that the line of curvature referred to inth e 

 preceding proposition is a principal section passing through 

 the extremities of the mean axis. Let U and U\ be its in- 

 terior umbilics, and U', U\ the umbilics diametrically oppo- 

 site. Geodetic lines drawn from U and Ui to a point 5, 

 taken anywhere in this principal section, make with the arc 

 UUy angles SUU\, SUiU, the product of the tangents of 

 whose halves is constant. Prolong either of these geodetics 



