Motion of an Electrified Sphere. 69 



provided that the double series remains convergent when the 

 modulus of every term is substituted for the term itself. This 

 will be the case if 



4>i! 2 sin 2 Ja(l + iii cos 2 J a) 1 

 (1 — n 2 | cos oil) 2 



where | cos a. | is the modulus of cos a. 



The quantity on the left increases with n^, and hence the 

 inequality will be satisfied if n x be less than the least real 

 value obtained from the equation 



4ft! 2 sin 2 \ot,(l-rn-i cos 2 ^oc) = (l — n 1 2 1 cos a | ) 2 . 



When « is not greater than -J-jr this equation leads to 

 (v2 — 1)*, or 0*643 ... as the value of n. When a is greater 

 than \it, the value is \/2 — 1, or 0*414.... Hence, if 

 n<v2-l the double series can certainly be rearranged as a 

 single series for all values of «. 



On rearranging this series, we obtain 



+ (105-189S 2 + 144S 4 -40S«) = -\- T) +. . . ."I, (45) 



where S = sin \a.. 



Mr. R. S. Cole has prepared Table VII., giving the values 

 of W/Uo for some values of n x and of a. 



W 



Table VII.— Values of 



Uo* 



n v 



a=0°. 



a = 80°. 



a = 60°. 



a = 90°. 



a = 120°. 



a = 150°. 



a = 180°. 



0, 







0-0018 



0-0067 



00134 



0-0201 



0-0250 



0-0268 



02 



o 



00074 



0-0276 



00547 



0-0818 



01013 



0-1084 



OS 







0-0177 



0-0650 



01281 



0-1895 



0-2333 



02492 



04 







0-0338 



01235 



0-2404 



03 .15 



04293 



0-4572 



0-5 







0-0588 



0-2118 



0-4050 



0-5325 



0-7038 



0-7465 



0-6 







0-0983 



03464 



0-6454 



0-9032 



1-0820 



1-1423 



07 







0-1653 



0-5623 



1-0087 



1-3779 



1-6125 



1-6922 



0-8 







0-2968 



0-9475 



1-6030 



21021 



2-4040 



2-5043 



0-85 







0-4221 



1-2773 



20704 



2-6433 



2-9804 



3-0911 



09 







0-6540 



1-8161 



2-7765 



3-4285 



3-8008 



3-9216 



