Electric Field in the neiglibourhood of a Spherical Bowl. 49 

 and 



as may be seen from the similar triangles, 



OBTj-OOyB', 



and 



OBT 2 ~OT 2 B'. 



a has the same signification as above. 



yfr and ijr' must be reckoned positive whenever the points 

 they are related to lie with on the same side of the plane of 

 the rim. 



Now equation (3) becomes 



y = ^_?Ji t an- 1 ±cR 



R 7r\R st cos {(j> — yjr) 



a . + cR'.OB I 



+ K'. OB tan a.s.tcos(<t> + ir-*)}' [ } 



The apparent want of symmetry of this formula with respect 

 to the points P and B vanishes when we consider that 



PB'.OB = P / B.OB, 



which relation is immediately proved by looking at fig. 3. 



§ 8. Formula (11) contains the general solution of the 

 second part of the problem mentioned at the head of this paper. 

 But to make it of general application, we have to explain 

 how to decide in each given case on the signs to be given to 

 tan 6 and tan Q' ; or, what amounts to the same, whether the 

 angles figuring in the equation are acute or obtuse. Now, in 

 our considerations on the potential of the isolated bowl, from 

 infinite space a finite part was singled out. For points within 

 that limited space the angle in the first term of the potential 

 was obtuse : the angle in the second term was obtuse when 

 the image of the considered point was situated within the space, 

 originally limited by the bowl and the plane of the rim, which 

 by inversion becomes a space bounded by the bowl and part 

 of the spherical surface that may be constructed through the 

 rim of the new bowl and the point P. Since in the original 

 figure the point at infinity is external to the space above 

 defined, in the transformed figure the point P (the image of 

 the point at infinity) is always external. 



Thus we have the following rule to decide whether the 

 angles in formula (11) are to be taken acute or obtuse ; in 

 other words, whether the upper or under sign is to be used. 



Phil. Mag. S. 5. Vol. 24. No. 146. July 1887. E 



