the Winding of Voltmeters. 103 



is the rate at which heat is produced in a layer of thickness 

 Br. 



This rate is — 2 r b ~ 2a times the volume of the layer. There- 

 to 

 fore the rate at which heat is produced per unit volume is 



C 2 ~r b ~ 2a : therefore, by hypothesis, the increase of tempera- 



2 Po v \ - 



2 



ture is proportional to C 5, — 2 r 



The increase of resistance of this layer is the resistance of 

 the layer multiplied by y times the increase of temperature, 

 and is therefore proportional to 



or to 



CV 



7_i_ r 1+26_4a 

 * 4 ' 



where y=y r c ~ b , and the whole increase of resistance is pro- 

 portional to 



I 



rX ry C 2f ^r l + b +°-* a dr, 



or 



CVo 2 



yo-4°Ol p -^),if^ = 2 + 6-4a + c; . . (7) 

 so that the fractional change of resistance is proportional to 



(rf-rf) (8) 



7o0 2 p 



Now regarding M as constant, and substituting for and 

 R from (6) (5) and (8) w r e observe that the fractional change 

 of resistance due to heating is proportional to 



^'W-^'n"-^' v ' ' ' {} 



and it is required to find the values of a, b, and c for which 

 this expression is a minimum. 



III. It may be seen that the error (9) is less as y p be- 

 comes less, therefore y p should be made as small as possible 

 so that when wires of two metals > as for example copper and 

 German silver, are employed, the wire for which yp has the least 

 value, that is the copper, should he used for the inner winding. 

 It is, however, when we consider the waste of electric energy 



