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



Therefore 



or 



2* lS2 *' 2 =8R 2 VpW( ? 2 + c 2 )fe 2 + c 2 ), 

 and 



[(s 1 +s 2 y-4a 2 y=SRHpp 1 (qq l +c 2 ) + VpViV+O fe 2 + c2 )}< 



And, because 



2a 

 tan (9= tanEPD: 



V(s,+s 2 ) 2 -4a 2 

 we have 



RV 



tan ^""2{ m (^ 1 + c 2 ) + <J?p x X? + <?)(q? + <?)}* ( 9 ) 



The radical must always be taken with the positive sign; 

 p and />!, analogous to q and ^ 1? have the same or the contrary 

 sign according as they fall on the same or the opposite side of 

 the plane on which they are normal. 



Equation (9) may be written in another form, and thereby 

 considerably simplified. To this effect we introduce the angle 

 APC, that we designate by 2(f), and the line PE = s. We see 

 easily that 



PE = s= V^ 



s x s 2 sin 2cf> = s 2 sin 2cf> = 2pc, 



s 2 ==sis 2 =2p \A/ + <2 2 , 



s^ cos 2<f> = s 2 cos 2<j> = 2pq* 



Now t and ty being quantities analogous to s and <f>, but 

 having reference to the point B, we find finally, by substitu- 

 tion in (9), 



2R 2 c 2 

 tan * & = ^ 2 cos2((/)-t)+6• 2 ^ 2, 

 or 



He 



tan0= + - TL T\ (10) 



st cos \(p — y) 



An expression of remarkable simplicity. 



§ 7. The same formula may be made use of to find an 

 expression for 6' . For R, t, and y]r must only be substituted 

 the quantities R' ? f 9 -v// which have the same relation to B', 

 the image of B, that R, t, and yjr have to B itself. 

 But (compare fig. 3) 



t x = t l= t l = 9^ == ^_ 



t x t 2 t "' a OB' 



