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surface of curvatures in the other principal planes are the one set 

 real, while the other are imaginary, as in the subjoined figure. 



It may easily be shown, as in the preceding figure, that in the 

 principal planes of the surface of the centres of curvature, the vertices 

 of the diameters of the evolute and ellipse are the vertices of the 

 ellipse and evolute in the adjoining plane. Thus the semiaxes OX, 

 Oa of the evolute and ellipse in the plane of XZ are the semiaxes of 

 the ellipse and evolute in the plane of XY. 



There are many other curious properties of this surface which will 

 be developed in the memoir. 



Before passing from this surface, I would mention that the funda- 

 mental property of the surface of centres suggests a simple property 

 of the evolute of an ellipse. 



