Mr. Louis Schwendler on Differential Galvanometers. 167 



resistances in the circuit ; and as w and w^ are given, <^ can only 

 be made a maximum with respect to g and ^', the resistances of 

 the two coils. 



Thus our physical problem is reduced to the following mathe- 

 matical one : — 



A function ^ containing two variables is to be made a maxi- 

 mum, while the two variables are connected with each other by 

 the relation 



A=y + ,,'_ P-^is + 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) {w^Jrg')-¥f{w + w'+g^-'g) ^ 2(g + w+f) 



P {9-^)9^ '^S^9'^9^—P{9^'^) 



* To some of the readers a more detailed working out of the mathema- 

 tical problem may perhaps be welcome ; and as this will also prove to be 

 an easy control over the equations (II.) and (IF.), I will give it here in a 

 somewhat condensed form. We had 



a°aK^A, . (I.) 



N 



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



resistances in the differential circuit ffig. 3), while d.=g'-{-w'---p ^-R.{gJ^w), 



N9 

 i. e. a resistance which at balance becomes =0; and further, 



^ = i9+w){9' + w')-]-fig-\-w-\-g'-^w'). 



DiflPerentiating a with respect to w', and remembering that — = 1, and sub- 



dw' 



stituting -— - = R, we have 

 aw 



da _r^j a/^_a RVff\. 



dw' I N N^ r 



.-. da = K(f)bw'. 



Thus the variation of a is always directly proportional to cf), a known func- 

 tion of g and g'; and to make 8a for any 8w' as large as possible, we have 

 to make cf) a maximum with respect to^ and g', while g and g' are connected 

 by the following equation : 



A=g'-i-w'-p^ig±w), (I.) 



p being a constant with respect to g and g', as also is A. 



We have therefore to deal here with a relative maximum ; and in accord- 

 ance with well-known rules we have to form the following partial differ- 



