46 Dr. J. N. Kruseman on the Potential of the 



radius, the following substitutions are necessary : — 



R% R 2 R 2 " 



»=-jjl V=yV, *=jU 



where f , tj, and f are the coordinates of the inverted point 

 sc y y 3 z y and p 2 stands for 



ew+e. 



The plane (4) becomes a sphere, 



(E-f)M'-f)H-f/=^ • <V> 



and the sphere (5) a plane, 



R 2 

 11% + vn + wZ— -£T (5 a) 



The point P remains unchanged, while B becomes the point 

 at infinity. The quantities to be found presently are: — 



(1) The radius a of the circle determined by (4a) and (5a). 



(2) The normal h from P on (5a). 



(3) The distance b from the centre of the circle to the foot- 

 point of h. 



Setting, for brevity's sake, 



cm + f$v + yw = P, 



we find without much difficulty, 



R 4 

 « 2 =VV*{V 2 -0>-P) 2 }, -.. (6) 



/i= w (R_2tt,) ' W 



+ {f + (W)|-E} S J 



or, after some reduction, 



i2= 4^-4^( 2 ^- PE ) +R2 - 5 F (8) 



It may be remarked that the perpendicular h will have the 

 positive or negative sign according as it lies with B on the 

 same or the opposite side of the plane. 



By these formulae, a, h, and b are expressed as dependent 

 on quantities artificially introduced by the use of a coordinate 

 system. We shall now proceed to get rid of those lines and 

 angles wholly alien to the problem, and to substitute for them 



