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co-ordinate axes in three points, the reciprocals of whose 

 distances from the centre are 5, v , I. 



"Let a, h, c, d, &c. be any constant quantities, and «, p, y, 3, 

 their reciprocals. If the general equation F (x, y, z, a, b, 

 c, dJ—0 be the symbolical representative of any property of 

 space or geometrical relation, the equation F (?, v, S,a,/3, r ,d)=0, 

 will give the reciprocal relation or the dual of the first. 



> are similarly involved and connected 

 S, v, ?, a, p, y , s, J 



in both cases by the same signs of operation. 



" By the aid of this very ramarkable theorem, we may reduce 

 the whole theory of reciprocal polars under the dominion of 

 analysis with the greatest ease ; the following are a few of the 

 most obvious subordinate relations deducible from this funda- 

 mental theorem. 



" Given the projective or tangential equations of any surface, 

 F (x, y, z,)=0, or <J> (§, v, S,)=0, we may write down the 

 tangential or projective equations of its reciprocal polar, 

 * (?, v, Z,)=Q, or * (x, y, ~J=0, from mere inspection. 



" Conceive a figure composed of points, right lines, planes, 

 curves of single and double curvature, and curved surfaces ; a 

 surface of the second order being described as directrix. 

 Imagine another figure constructed, whose points, right lines, 

 and planes shall be the poles, conjugate polars, and polar 

 planes, of the planes, right lines, and points, of the original 

 figure ; these two figures may be called reciprocal polars, one 

 of the other. 



" From the reciprocal relations between the two equations 

 *(?, v, 2,)=0, and * (.?•, y } sr,)=0> we may conclude that 

 if— 



" In one figure, a surface passes through n given points, in 

 the other we shall have a surface touching n given planes. 



" In one figure, a group of right lines being situated in the 

 same plane, in the other we shall have as many right hues 

 meeting in a point. 



