﻿and on the Gravity-Potential of a Ring. 469 



[nside, 



87rRcos 3 aN2 S 3 



* - 3 sin 2 a ' (O-C) 8 



where «„, /3 H arc the expressions given above. 



When a (or k) is small, the values can be expressed in a 

 Pew terms of a series of powers of k as iii the former i 



These value- are as follows : — 



h-j5-(i + 1 » 



105 L6 



<* 2 = j2g^r* 



A .J{l + |i.-gL-")».}, 



a- I- 



When A; is not large, the integrals in the expression for 

 the self-induction may be found approximately in a series 

 of powers of k. If this be done, after some rather tedious 

 calculations the value is found to be 



. t> T 1 2 COS 2 a + 3 COS 3 a + 4 COS 4 a 

 47rK 



+ 7 



L 6 3 (l + COSa) :J 



w. { 6L+ , +(39| ,_ 5)il( ^ L _™ >l}] 



3(1+ cos 



When & is very small this is 7rR(4L — 7). 



The value of yfr' will give the coefficient of mutual induc- 

 tion between the circular coil and a thin circular wire with 

 the same straight axis and with its plane parallel to the coil. 



