102 Professors Ayrton and Perry on 



fine wire coil of an Ohm-meter, Powermeter, and Ergmeter, 

 or indeed in any case where it is desired to produce a given 

 magnetic effect with a minimum error due to heating. It is 

 only necessary to consider the case of a cylindrical coil of 

 internal radius r and external r x and length I, as it is easy 

 to extend our reasoning to a coil of any other shape and 

 dimensions. 



IT. At a place in the coil at a distance r from the axis let 

 the cross section of the wire be x, let p be the specific resis- 

 tance of the material, and 7 the rate at which p increases with 

 temperature; x, p, and y being functions of r. For simplicity 

 of calculation we shall neglect the volume of the insulating 

 material. Let one spire of radius r produce a magnetic 



effect of the amount rrr{ /1X 



J&Xird, (1) 



C being the current, and d having in almost all instruments 



a value which lies between the limits — 1 and 0. Also let 



x=x r* (2) 



P=por b (3) 



P r Y = PoVor c ....... (4) 



I Br 

 In a layer of wire of thickness Br there are spires of 



wire, and the resistance of the layer is 



2nrrpl ~ 27rrp rH ~ 27rp lr l + b - 2a . 



f-or, or — fr-or, or ru 9 Br; 



x z 7 # V« 7 x\ 



so that the whole resistance of the coil is evidently 



R=^ (V*-^) if m = 2 + b-2a. . . (5) 

 mx* v / v 7 



The magnetic effect of the spires in the thickness Br is 



KCr d l— or r d ~ a Br, 



x x 7 



so that the whole magnetic effect of the coil is 



m=^ov'-»- ») if »=<*-« +1. • • ( 6 > 



We shall now assume that we can, by introducing thin strips 

 of copper among the windings, give the same ease of cooling 

 to unit volume of the wire everywhere ; that is, after the cur- 

 rent has flowed for some time, that the increase of temperature 

 at any point is proportional to the rate at which current heat 

 is produced per unit volume. Now 



# X Q 



