Perkins — Methods of Using the Galvanometer. 55 



To compare S x and S 2 , the most obvious method is to find 

 what conditions render them equal. Setting S> = S 2 we have 

 r 5 3 — 4r a 2 1\ — 4r* = 0. The solution of this cubic gives 

 r b = 2*4 ?\, approximately. A further comparison indicates 

 that if the galvanometer resistance is less than 2'4 times that 

 of the cell, the series arrangement is best, although when the 

 galvanometer resistance is zero they are again equal. All values 

 of r b greater than 2*4 ?\ give S x greater than S 2 , and when 

 r & = 6/*! the bridge method is twice as sensitive. 



The following table gives an approximate idea of the relative 

 value of the methods. E is assumed = 100 

 and r 1 = 1. The values are given to the 

 nearest integer. 



Another aspect of the problem arises when 

 a certain maximum current through the varia- 

 ble resistance is admissible. Let this current 

 = K. Then in case of the bridge method 



K = 



2K 



substituting for E in S x we have 

 very nearly. In order to obtain 



r 5 







S 2 



s, 



100 



100 



r l 



25 



17 



2r 1 



11 



10 



3r i 



6 



7 



4r 1 



4 



6 



5r i 



3 



5 



6r a 



2 



4 



1r l 



1-6 



3'4 



3^ + 4r o 

 this simple value it was assumed that in all 

 cases where the current through r x arm of 

 the bridge is limited, r x will be considerably 

 smaller than r b ; and even if i\ is half as large as ?' 5 , the error 

 is still very small. 



Now S 9 under these conditions = hence it is clear 



r 4- r 



' 5 ~ ' 1 



that the series method is always best when the current in r 1 

 must not rise above K. 



One more case remains ; when the allowable current through 

 the galvanometer is less than that allowed through the varia- 

 ble resistance. In this case S x becomes equal to S 2 when 

 K' _ 2(r 6 + r,) 



when E? 



is the maximum 

 K' 



current allowed 



K 4r 5 + 3r i 



K' 



through the galvanometer. If — is smaller than this, the 



bridge method is clearly best. When K / is very small indeed 

 as compared to K this method is vastly more sensitive than 

 the other. 



Trinity College, May, 1904. 



