Dr. Hirst on Equally Attracting Bodies. 165 



y,«'0, therefore, is a right angle, and the locus of m< a sphere on 

 Op' as diameter. , r <. c 



As the plane recedes from or approaches O, the diameter ot 

 its inverse sphere diminishes or increases, so that the centre of 

 inversion itself must be regarded as the inverse of the mhnitely 

 distant plane, and a plane through the point of mversion as co- 

 inciding with its own inverse. The converse of the present 

 theorem needs no demonstration ; from the two we conclude, 

 too, the following and its converse : — 



III. The inverse of a line is a circle, passmg thiough the cen- 

 tre of inversion, whose plane coincides with that determined by 

 this point and the line, and whose centre is upon the perpendi- 

 cular let fall from the former upon the latter. 



IV. The inverse of a sphere is also a sphere. 



Let S be the centre and s the radius of the given sphere; then 

 if Si be the inverse of O with respect to the given sphere (b), 

 regarded as a sphere of inversion, we shall have 



SO. SS"i = s^^ = const., 

 and every circle described through and Sj will cut the given 

 sphere perpendicularly. Now, by III., the inverses of all such 

 circles constitute a pencil of rays through the point S', the niverse 

 of S. and from I. it follows that every such ray cuts the inverse 

 of the given sphere perpendicularly, so that this inverse must be 

 itself a sphere around S'l as centre. The centres of the two 

 spheres are corresponding, but not inverse points, for they are, 

 clearly, connected by the relation 



80 that the centres S and S', are on the same side or on oppo- 

 site sides of O according as the latter point is without or within 

 either sphere ; in all cases, however, is a centre of similitude 

 of the inverse spheres. When either cuts the sphere of inver- 

 sion orthogonally it coincides with its own inverse, for then 

 g2_os^_s2_ When the given sphere (S) passes through O, 

 the latter point and S, coincide, so that S', recedes to infinity, 

 the rays through the same become parallel to OS, and the sphere 

 (S'l) cutting these rays perpendicularly degenerates into a plane 

 perpendicular to OS, in accordance with the converse of II. 



V. The inverse of a circle is also a circle. 



For the first circle may be regarded as the intersection of two 

 spheres whose inverses, by IV., being also spheres, must intersect 

 in a second circle, the inverse of the former. It is evident that 

 tbe two circles lie on a sphere which cuts the sphere of inversion 

 orthogonally, and that they are at the same time anti-parallel 



