THE MEASUREMENT OF CURRENT 15 



the wire has been so wound that the boundary of the cross-section 



of t he coil is given by r 2 = sin a, where e v has some particular 



e v 



numerical value. If an attempt be then made to alter the form 

 of the cross-section by changing the position of a portion of the 

 wire, the field at the needle will be reduced, for the only change 

 p<.<sible is that to a region of less efficacy. Consequently, the 

 equal ion is t hat of the boundary of the best form of cross-section. 

 The curve is symmetrical about the maximum radius vector, the 



value of which is - . 

 Ve v 



Similarly, to arrange a given number of turns to give the maxi- 

 mum value of G, their traces must be included within the curve, 



r = ( ) sin 2 a. The gain from using the best form of cross- 



\ f I 



section is small. 



Graded Coils. As the inner turns of the coil of a Thomson 

 galvanometer are very near the needle they are much more effect- 

 i\v than those in the outer portion of the coil, which, while they 

 add much to the resistance of the instrument, contribute com- 

 paratively little to the galvanometer constant. This suggests 

 that with a coil of a definite resistance, it might be best to con- 

 centrate the resistance in the turns near the needle, winding this 

 part of the coil with a finer wire than that used for the outer 

 portion. Maxwell has shown in his "Treatise on Electricity and 

 Magnetism," * Art. 719, that the diameter of the wire should 

 increase with the diameter of the layer of which it forms a part, 

 the exact law depending on the relation between the diameters 

 of the covered and the bare wire. 



As it is not possible to wind the coil with a wire having a cross- 

 section which is a function of its distance from the end of the 

 wire, it is customary to wind the coil in three or four sections, each 

 of a single size of wire. 



In a very sensitive galvanometer having a resistance of 25 

 ohms, the three sections were wound with No. 40, No. 34 and 

 No. 26 B. & S. gage. The thickness of the insulation was 0.002 

 cm. The galvanometer constant with the instrument so wound 

 was about 33 per cent, greater than if a No. 26 wire had been 



* Third edition. 



