478 Prof. A. Gray on Relations of 



But we may also, since by fig. 4 r(d<p + dd) jets' = cos 0, 



write 



C' cos 0ds' _ C"d± Cdd 

 S r ' Jo r + J r ' 



. (17) 



Fijr. 4. 



The first integral is obtained at on 



ce. 



C«d$_ 



a + a 



,K(A), 



t comes out 



(18) 



am 



it is to be observed that multiplied by 2p'a' it is the 

 potential at A due to a uniform distribution of electricity 

 (or gravitating matter) of line density // on the outer circle. 

 For the second integral idOjr we take 



j that is 



a 2 — a' 2 -j- r 2 — 2a' r cos #, 

 = a cos 6 ± (a 2 — a' 2 sin 2 0)K 



• (19) 



Thus the largest value of 6 is sin _1 (a/a'), and is shown 

 by the dotted lines in fig. 4, where a right angle at A is 

 intended to be indicated. As successive positions of E and 

 E' are taken on the semicircles, AEB and PE'Q, 6 varies 

 from zero to sm _1 (a/a') and then falls off to zero. 



For the first part of this range of variation we have to 

 take the lower sign in the last equation, and for the second 

 part the upper sign. Thus we get for the first part, putting 



