406 On Electro-Magnetic Rotation. [Mar. 30, 



But if through any local circumstance, as in the experiment in air, or 

 through heating of the chamber as in the exhausted tube, or otherwise, 

 the path of the current be distorted and displaced, then each element will 

 be subject to the action of two forces. To estimate these, let ds be an 

 element of the path, with rectangular components dx, dy, dz, C the 

 strength of the current, and B the magnetic force with components X,T, Z, 

 w r hich in the first instance will be treated generally. Then one force 

 will be that tendiug to impel the current in the direction of the axes 

 respectively, and may be expressed by 



C(Ydz-Zdy):d$, C(Zdx-Xdz) :ds, C(Kdy-Ydx) : ds. 



Besides this, there will be the tendency of the current to follow the 

 shortest path so as to diminish the resistance. Bepresenting this as % 

 tension r, the components at one end of ds will be 



— rdoo : ds, — rdy : ds, — rdz : ds, 



and those at the other 



(rdx : ds) + d(jdx ids) ... , 



the algebraical sums of which are 



d(jdx : ds), d{rdy : ds), d(rdz : ds), 



and the equations of equilibrium then become 



C(Ydz-Zdy) +d( T dx:ds) = 0, (1) 



C(Zdx -Xdz)-\-d(rdy:ds)=0, (2) 



C(Xdy-Ydx) + d(rdz:ds)=0; ..... (3) 



taking s as the independent variable and multiplying by dx : ds, dy : ds, dz : ds 

 respectively, and adding, we obtain dr=0, or r = constant. Again, mul- 

 tiplying by X, T, Z and adding we obtain 



Xd 2 x:ds 2 +Yd 2 y:ds 2 + Zd 2 z:ds 2 =:0, .... (4) 



which expresses that the absolute normal (or normal in the osculating 

 plane) is perpendicular to the resultant magnetic force. 



In the case of a uniform tint, X, T, Z will be constant. Integratiug 

 (4) and putting i for the angle between the tangent and the lines of 

 magnetic force, we find 



X.dx + Ydy + Zdz= Et?5 cos i, 

 so that the tangent line is inclined at a constant angle to the line joining 

 the poles. 



Again, the following combinations, (2) dz-(3)dy— 0, (3) dx— (1)^=0, 

 (l)dy~(2)dx^0 give 



or 



