1S72.] Louis Schwendler — On Differential Galvanometers. 140 



«°aK^A 



N 



Differentiating this expression with respect to w ', the external resistance 

 belonging to the coil g ', we get 



da = K ( <Jg __ A E x/g ) 



dw> { N N 2 j 



where K = 



dw' 

 or the variation of the deflection a, when to ' varies, is 



( N N 2 ) ^ 



Now it is clear that the instrument is most sensitively constructed 

 when, for the slightest variation in w ', the variation in a is greatest. This 



will be the case if the factor cj> = ^rr ■ — ' — AT2 - is as great as possible. 



This factor <j> is a known function of the resistances in the circuit, and as 

 w and to ' are given, cf> can only be made a maximum with respect to g and 

 g ', the resistances of the two coils. 



Thus our physical problem is reduced to the following mathematical 

 one : 



A function <£ containing two variables is to be made a maximum, while 

 the two variables are fixed to each other by the relation 



A = g 1 + to' — p —¥- (g -f- w), 



A being a constant with respect to g and g ' and becoming zero at balance. 



Solving this question (relative maxima), we get 

 (w— g) (to< + g')+f(w + io> + g> — g) _ 2 (g + to +/) 



P (y — w)3' 2 ^/g x/g' — p (g + w) 



..II.' 



* To some of the readers, a more detailed working out of the mathematical pro- 

 blem may, perhaps, be welcome ; and as this will also prove to be an easy control 

 over the equations (II) and (II'), I will give it here in a somewhat condensed form. 

 We had 



\Jg 

 a a K ■ A I 



where K represents a constant, i. e. a quantity independent of any of the resistances 



in the differential circuit (Fig. 3), while A = <r* -J- w* — p %JL (g -J- w), i. c. a r'e- 



s/'J 

 sistance which at balance becomes = ; and further 



n=(j + «) {>/ H-w') + / (o + ™ +g' + w'}. 



Differentiating a with respect to w' f and remembering that - — = 1, and substitut- 

 ed ' 



dN „ . 



mg = It, we have 



& dw* 



20 



