ON AN ELLIPSOID. 89 



These values belong to the umbilici of the ellipsoid ; a result 

 easily anticipated. When they are introduced in (9), it becomes 



2 2 



2 = 0, and similarly j* = 0. 



Hence (8) reduces to 



i> = (16), 



and represents the principal section of the ellipsoid, which passes 

 through the greatest and least axes. In this case then, as our 

 analysis would lead us to anticipate, the lines of curvature 

 coincide; a result which, although well known, seems not very 

 accurately demonstrated by Leroy. After having shown (p. 309 

 of the second edition) that the two directions of curvature 

 coincide at the umbilical points, he proceeds to integrate, and 



passes from ^ = to y = ^, and thence, determining the con- 

 stant, to y ; which last represents the line of curvature 

 sought. But -j- has been shown to have the value 0, only for 



u/ZC 



the umbilical points, and we are therefore not at liberty to pass 

 by integration from these to any other points at which this 

 may not hold. Were the process legitimate, it would lead to 

 the strange conclusion, that the lines of curvature through an 

 umbilicus are necessarily plane curves. 



As there appears to be still some difficulty with regard to 

 the theory of these singular points, we may enquire whether, 

 in order to determine the lines of curvature through any point 

 whatever, more is requisite than to substitute its co-ordinates in 

 the general equation of the lines of curvature, and thus to get 

 two values for the arbitrary constant; whether the result can 

 ever be indeterminate, except when the lines, as at the extremity 

 of an axis of revolution, are so in reality. In this view we see 

 at once, that the process given by Leroy after Poisson for de- 

 termining the directions of curvature at an umbilicus, is simply 

 the ordinary method for ascertaining the position of the branches 

 of any curve at a multiple point ; and that the result arrived at, 

 is not that more than two lines of curvature pass through an 

 umbilicus, but that every point which, with reference to the 

 surface, is umbilical, is, with reference to the lines of curvature, 

 a multiple, or more generally a singular point. These sug- 



