537 



by which 1 arrived at it, as it involves a principle of very- 

 useful application in the theory of reciprocal polars. 



" Given a point and a plane, — if we take the reciprocal 

 plane and point with regard to any sphere, the ratio of the dis- 

 tances of the first point from the centre of the sphere, and from 

 the first plane, is equal to the ratio of the distances of the 

 second point from the centre of the sphere, and from the second 

 plane. 



" Hence, since in a sphere the distance of any tangent 

 plane from the centre is constant, the reciprocal of a sphere is a 

 surface such that the distance of any point from a fixed point 

 is to its distance from a fixed plane in a given ratio. 



" Again, since in an ellipsoid of revolution round the axis 

 major the product of the distances of any tangent plane from 

 the foci is constant, the reciprocal of such an ellipsoid is 

 a surface such, that the square of the distance of any point 

 from a fixed point is in a constant ratio to the product of its 

 distances from two fixed planes. Or, more generally, if a sur- 

 face be such, that the product of the distances of a tangent 

 plane from n fixed points is constant, the reciprocal surface will 

 be such, that the ratio of the wth power of the distance of any 

 point from a fixed point, to the product of its distances from /* 

 fixed planes, will be constant. 



I add one or two instances of transformations of plane 

 curves by the same principle. 



In a conic section the product of the distances of any point 

 on the curve from two fixed tangents, is in a constant ratio 

 to the square of the line joining their points of contact. 

 Hence, the square of the distance of any tangent to a conic 

 section from a fixed point, is in a constant ratio to the product 

 of its distances from the two points of contact of tangents 

 drawn from the fixed point. 



" As a particular case of this, we derive the well-known 

 property, — any tangent to a conic will intercept, on two fixed 

 parallel tangents, segments whose rectangle will be constant. 



