181 



Hence the sections of the surface of centres in the principal planes 

 are two in each ; one an ellipse, the other the evolute of an ellipse. 



On the umbilical lines of Curvature. 



Among the French mathematicians there has been much difference 

 of opinion as to the nature of the lines of curvature which pass 

 through the umbilicus of the ellipsoid. Some hold with Monge and 

 Dupin, that the two lines of curvature which everywhere else on the 

 surface are at right angles to each other, here merge into one. This 

 is such a violation of the law of continuity, that others adhere to the 

 opinion of Poisson and Leroy, to the effect that at the umbilicus the 

 radii of curvature are all equal, and that there is an infinite number 

 of rectangular systems of lines of equal curvature all passing through 

 the umbilicus. 



An examination of the surface of centres will demonstratively show 

 that the latter opinion is the correct one. 



For this purpose let a tangent plane to the surface of centres be 

 drawn through the umbilical normal. Now the protective coordinates 

 of the umbilicus are 



(10) 



a c a c 



and the segments of the axes of x and z cut off by the normal are 



ca- ( 



c a 



Hence the tangential equations of the normal in the plane of xz are 



<> - 



Now substituting these values of and in the equation (9) of the 

 surface of centres, we shall have for the value of v 2 the following ex- 

 pression : 



[(a 2 - 



or v= . 







Hence an infinite number of tangent planes may be drawn through 

 the umbilical normal to the surface of centres. 



The principal sections of the surface of centres in the mean plane, 

 or in the plane of xz, the plane of the greatest and least axe, possess 

 some very curious properties. 



