used in Electrical Measuring-Instruments. 435 



instruments as are in common use, has been almost entirely 

 ignored. 



In a paper "On Galvanometers," by Prof. Ayrton, Dr. 

 Sumpner, and the writer, read before the Society on Jan- 

 uary 17th, it was pointed out that the coils of d'Arsonval 

 galvanometers should be narrow and long, and that there 

 should be no internal core. Mr. C. V. Boys had evidently 

 arrived at the same conclusion long before the date referred 

 to, for the coil of his radio-micrometer is a proof of this. 



The object of this paper is to determine the best shape of 

 the section of the coil perpendicular to the axis about which 

 it turns. 



The subject will be dealt with as concerning coils suspended 

 in uniform fields, but similar reasoning may be applied to 

 other instruments in which movable coils are used. 



Let the point (fig. 1) represent in plan Fig. 1. 

 the axis about which the coil turns, and assume A- 

 it is placed in a magnetic field whose direction j 

 is perpendicular to A B. ^ 



Let P be an element of the section of the 

 coil, then the deflecting moment exerted by unit 

 length measured at right angles to the paper is ° 



HCar sin 0; .... (1) 



i 

 where H is the strength of field, C the cur- 

 rent-density per unit area, a the area of the ! 

 element, and r its distance from the axis. B 



The moment of inertia of the element about will be 



war 2 , ........ (2) 



where w is the mass per unit cube. 



Now in ordinary commercial instruments it is important 

 that the period of oscillation should not be inconveniently 

 long, and that the power consumed by the instrument should 

 be as small as possible ; then, since for a constant period the 

 controlling moment at unit angle must be proportional to the 

 moment of inertia, the problem resolves itself into finding the 

 shape of the section such that the total deflecting moment for 

 a given total moment of inertia is a maximum. 



The ratio of the Deflecting Moment to the Moment of 

 Inertia for the element above considered is 



HCtfr sin 6 . sin B 



2 — , i. e. " -j 



since H, C, and w may be considered constants. 



