78 HENRY A. KOWLAND 



temper; 2d, the permanent magnets must be magnetized to the same degree 

 at similar points of the systems; 3d, the coils of the electro-magnets and 

 other wires or bundles of wires carrying the current must have similar 

 external dimensions in the two systems and must have the product of the 

 current by the number of wires passing through similar sections of the two 

 systems proportional to the linear dimensions of the systems. 



This will apply to the case we are considering when the product of 

 the current by the number of the turns of wire varies in direct propor- 

 tion to the size of the apparatus. Hence in this case \ and !-i f 



dx ay 



will vary inversely as m. Hence we see that n will be inversely pro- 

 portional to the size of the apparatus; and although we have only 

 proved this for the case when * is small, it is easy to see that it is 

 perfectly general. The advantage of small diamagnetic apparatus is 

 thus apparent, for the smaller we make it the more vibrations the bar 

 will make in a given time and the more promptly will the results be 

 shown. 



It might be thought that by hanging a very small bar in the field oi' 

 a large magnet, we might obtain just as many vibrations as by the use 

 of a small apparatus; but this is not so, for Sir Wm. Thomson has 

 shown 2 that the number of oscillations of a feebly magnetic or diamag- 

 netic body of elongated form in a magnetic field is nearly independent 

 of the length when that is short. So that the only way of increasing 

 the number of vibrations is to decrease the size of the whole apparatus, 

 or to increase the power of the magnets; the latter has a limit and 

 hence we become dependent on the former. 



The theory of the effect of the size of the body is very simple, and we 

 may proceed as follows. Let the body be in the form of a small bar 

 whose sectional area, a, is very small compared with its length, and let 

 f be the angle of the axis of the bar with the line joining the poles, and 

 r the radius vector from the origin. Developing R 2 as a function of 

 x and y by Taylor's theorem, and noting that as R is symmetrical with 

 reference to the planes XZ and YZ, only the even powers of x and y 

 can enter into the development, we have, calling R the value of R 

 at the origin, 



2 \ dy? dy 



r#(/2n 



2.3.4V dtf dtfdf dy* 



2 Reprint of Papers, art. 670. Remarques sur les oscillations d'aiguilles non crys- 

 tallisees. 



