537 
by which I 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 x fixed points is constant, the reciprocal surface will 
be such, that the ratio of the nth 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 
 eurves 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. 
_ * Asa 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. 
