126 



" In the one, points being common to two or more curves of 

 double curvature ; in the other, so many tangent planes exist 

 to two or more developable surfaces. 



" In the one, if a series of tangent planes may be drawn to 

 a curved surface through a point on it ; in the other, we shall 

 have as many points of contact of a curved surface with a tan- 

 gent plane. Hence, if a cusp is found on the one, a curve of 

 plane contact exists on the other. 



" As the reciprocal polar of a surface of the second order 

 is also a surface of the same degree, a great variety of the 

 properties of such surfaces may be deduced in this manner ; 

 for example, it may easily be shown, that if two surfaces of the 

 second order intersect in a plane curve, they must again inter- 

 sect in another plane curve j hence, if two such surfaces are 

 enveloped by one cone, they are also enveloped by a second. 

 And thus a duality exists between the properties of curves 

 and surfaces of the second degree, which in the general case is 

 found only between curves and surfaces, and their reciprocal 

 polars. Again — 



" If one surface is generated by rectilinear generatrices, so 

 also will its reciprocal polar ; hence, of surfaces of the second 

 order, the hyperbolic paraboloid, and the continuous hypcrbo- 

 loid, or that of one sheet, are exclusively reciprocal polars of 

 themselves or of each other, whence it follows, that the pro- 

 perties of those surfaces are reciprocal to each other. 



" On a future occasion I hope to have the honour of laying 

 before this Society some applications of this method, in the 

 theory of terrestrial gravitation, and in that of the rotation of 

 a rigid body round a fixed point. 



