224-227J 



Inversion 



199 



Corresponding to any point P we can find a second point P', the inverse 

 to P in the sphere. These two points are on the same radius at distances 

 from such that OP. OP' = K\ 



As P describes any surface PQ..., P' will describe some other surface 

 P'Q 7 . . . , each point Q' on the second surface being the inverse of some point 

 Q on the original surface. This second surface is said to be the inverse 

 of the original surface, and the process of deducing the second surface from 

 the first is described as inverting the first surface. 



It is clear that if P'Q ... is the inverse of PQ..., then the inverse of 

 P'Q'... will be PQ.... 



If the polar equation of a surface referred to the centre of inversion 

 as origin be f(r, 6, <) = 0, then the equation of its inverse will be 



f( , 6, <f>] = 0. For the polar equation of the inverse surface is by 

 definition / (/, 0, <f>) = 0, where rr' = X 2 for all values of 6 and 0. 



Inverse of a sphere. Let chords PP', QQ', ... of a sphere meet in 

 (fig. 68). Then 



OP.OP'=OQ.OQ'=... = * 2 , 



where t is the length of the tangent from to the sphere. Thus, if t is the 

 radius of inversion, the surface PQ... is the inverse of P'Q'..., i.e. the sphere 



P' 



FIG. 68. 



is its own inverse. With some other radius of inversion K, let P"Q" ... be 

 the inverse of PQ . . . , then 



sothat 



OP" OQ" K 2 



__ = __ = ... = ._ 



and the locus of P", Q", ... is seen to be a sphere. Thus the inverse of a 

 sphere is always another sphere. 



