264 Mr. Louis Schwendler on Differential Galvanometers. 



the total resistance in the battery branch, and p an absolute 

 number expressing what was termed the " mechanical arrange- 

 ment" of the differential galvanometer under consideration. 



By these three equations, which are independent of each 

 other, g, g f , andjj can be expressed in terms of w, w' } and/. 



By equation (I.) we have, at or very near balance, 



a' + w' Vq 



g + w Vg 



which value, substituted in equations (II.) and (II'.), gives 



{w-9)W+g')+f{w+v!+tf-g) = Wg+w+f) ( 



(g'+w'Kg-^g' W-d)(g+w) " ^ ' J 



and 



{g + w){£f—v})g {g-w){g l + ™y l ' ; 



and from these two equations g and g 1 may be developed. 



This is best done by subtracting equation (II.) from equation 

 (11/), when, after reduction, we get 



(w'g — wg') (w ! g + wg' +gg ! + ww 1 ) 



^-fig+g' + w + io'jiiu'g-iug 1 ). . . (III.) 



Now it must be remembered that, with respect to our physical 

 problem, /, w, w\ g, and g' represent nothing else but electrical 

 resistances, and that they have therefore to be taken in any 

 formula as quantities of the same sign (say positive). 



Consequently the above equation (III.) would contain a ma- 

 thematical impossibility (a positive quantity equal to a negative 

 quantity) whenever the common factor w'g—wg'is different from 

 zero. 



In other words, equation (III.) can only be -fulfilled if we 

 always have 



v/g—tqgf=0 (IV.) 



This simple relation between the resistances at which balance 

 arrives and the resistances of the two differential coils expresses 

 not only the necessary and sufficient condition under which a 

 simultaneous maximum sensitiveness can exist, but it also affords 

 an easy means of getting at once those special values of g, g ! , 

 aftdp, which only solve the physical problem. 



Substituting the value of either g or g 1 , as given by equation 

 (IV.), in s equations (II.) and (II'.) and developing g and g\ we 

 have 



