446 Dr. W. F. G. Swarm on the Motion of the 



entered the quadrant. F(£) is of course proportional to the 

 potential of the quadrant. 



There are two cases of common occurrence. In the first 

 case the difference of potential driving the electricity into 

 the quadrant is large compared with the rise o£ potential of 

 the quadrant during the observations, in which case F(£) is 

 of the form X£, where X is a constant. The second case to 

 be considered is the one in which the alteration of the current 

 due to the rise of potential of the electrometer is of im- 

 portance. In this case if V is the potential of the quadrant, 



dY . 



—7- is of the form A — AY, where A and h are constants, so 

 eft 



that 



Y=Y 1 (l-e- ht ), 



Y 1 being the final potential which would be attained by the 

 quadrant after an infinite time. Since ~F(t) is proportional 

 to V, it is of the form P(l — e~ ht ) where P is a constant. 



The solution of (1) corresponding to the case where F(t) 

 is of the form Xtf is 



Q X r XT -d ™ . /2tt O 



where X, the logarithmic decrement, =5T/4K, and 



2tt_ f a _ b 2 

 T " V K 4K 2 * 



On introducing the conditions = when t = 0, and 6 = 

 when t = 0, we readily obtain 



, 2ttX T 



tan 7 =-5 — r-s, &=•—. 



1 7T 2 — X 2 ' 4"7T 



IIE the needle were allowed to come to rest for various 



values of the potential of the quadrant, the position it would 



take up would of course depend upon that potential. We 



shall speak of the " Ideal Motion " of the needle as the 



motion it would have if, for each value of the potential of 



the quadrant, it assumed the same position as in the above 



statical case. For the ideal motion corresponding to the 



Xi X£ 

 present case we have 0= — . We shall call the 



x . a a 



error. By differentiating the error with respect to t and 

 equating to zero, we find that it is a maximum when 



(¥-*)= 



= r- 



