Mr. Louis Sehwendler on Differential Galvanometers. 267 



N-2 — 

 da _ ^ dg _ U 



^~ 2y^N 2 ~~~V 



and 



V— -U — 

 J 2 «_ dg dg 



df~ W ; 



but . • » ■ 



d 9 

 it follows that U = 0; 



d?a_ 1 d\J 

 *'• dg*~Ydg' 

 Now 



dU _ _/tfN d*N\ 



dg" \dg + ' 9 dg*)> 



dN d 2 N 



but —7-, as well as -j^, being invariably positive, it follows that 



- r - is invariably negative ; and as, further, V is always positive, 



d 9 d* 2 a . 



it follows finally that -^ is always negative, or the value of g 



obtained by equation -j- =0 corresponds to a maximum sensi- 

 ag 



tiveness of the differential galvanometer. 



In a similar way it can be shown that the value of gf obtained 



by equation -p =0 corresponds also to a maximum sensitive- 

 ness of the differential galvanometer. 



This is in fact a second and far more simple solution of the 

 problem. However, it is by no means as general, nor does it 

 adhere as closely to the spirit of analysis, as the first more com- 

 plicated solution. 



Effect of Shunts. — It is clear that the introduction of shunts 

 cannot alter the general results as given in equations (a), {b) } 

 (c)j and (d), as long as the shunts are used merely for the pur- 

 pose of carrying off a fixed quantity of current without in them- 

 selves having any direct magnetic action on the needle. 



However, to avoid misunderstanding, it is well to remember 

 that, in the case of shunts being used, the values to be given to 

 w and w' in the above equations are not those at which balance 

 actually arrives, but those at which balance would arrive if no 

 shunts were used ; i. e. the resistance at which balance is esta- 



