Mr. Stubbs on a new Method in Geometry. 345 



of this cone with the surface inverse to the ellipsoid is the 

 line of curvature. 



5. By putting x* + y* -f s 2 = a constant, we find — g- 



or 



tf , * 



+ tb-H — 5- = const., or the intersection of the surface of 

 o* c* 



elasticity and concentric sphere is a spherical conic, since 

 it is the same as the intersection of the surface of second 

 order and sphere. 



6. The circular sections of the surface of elasticity corre- 

 spond to those of the ellipsoid, and the umbilici of either are 

 found by the intersection of the diameters conjugate to the 

 circular section of the ellipsoid, as at the umbilici the ultimate 

 section must be a circle, and therefore parallel to the circular 

 sections. 



7. As to the rectilinear generatrices of the hyperboloid of 

 one sheet correspond circular sections in the inverse hyper- 

 boloid, the latter has an infinite number of circular sections 

 passing through the centre, but only two whose centre is at 

 that point. 



8. Hence the inverse hyperboloid of one sheet may be de- 

 scribed by a moveable circle passing through a given point 

 and moving on three others passing through the same point, 

 which only cut in that point, and which neither lie in the same 

 plane nor are circles of same sphere. 



9. From the property that the sum of the squares of the reci- 

 procals of three radii vectores at right angles to each other is 

 constant in the ellipsoid, it follows that the sum of the squares 

 of three rectangular semidiameters in the surface of elasticity 

 is constant. 



10. As the locus of the feet of perpendiculars from the cen- 

 tre on tangent planes of an ellipsoid is a surface of elasticity ; 

 by inversion, the locus of centres of spheres tangent to a sur- 

 face of elasticity and passing through the centre is an ellipsoid. 



11. From 8 and 5 it follows, that if planes pass through 

 the centre of a surface of elasticity and cut out sections of a 

 constant area, they envelope a cone of the second order, since 

 the sum of the squares of the axes of the section is constant. 



The foregoing are a few of the general theorems that may 

 be deduced by the method I propose ; they furnish a new in- 

 stance of the duality that Chasles and others have remarked 

 between the properties of figures ; but it is superior to any 

 hitherto proposed, as we can by it arrive at once at the properties 

 of curves of higher orders which surpass bur present power of 

 analysis, from those of known curves; as from the known pro- 



