266 Mr. W. Williams on the Relation of Dimensions 



5. Current density =C = C (YZJ-^YZ-'T -1 = Angular 

 velocity. The velocity is instantaneously directed along Y, 

 the magnetic axis, and the axis of rotation is X, the axis of 

 the electrical displacement. If the angular velocity be that 

 of a vortex filament coinciding with the current, the strength 

 of the current, C, becomes the strength of the vortex — product 

 of angular velocity into cross-section of filament. 



6. Electric displacement = D=CT = YZ _1 = an angular dis- 

 placement. [According to the elastic solid theory, D would 

 be of the dimensions of a shear, the planes XY being displaced 

 parallel to each other in the direction Y.] 



7. Electrical force = E. Since [ED] =MY(XZ) _1 T" 2 ,and 

 [D]=YZ _1 , E = MX~ 1 T" 2 . Taking D as an angular dis- 

 placement, E must be of the dimensions of a torque, for ED 

 is of the dimensions of work done per unit volume. [Let a 

 cylinder rotate in a resisting medium. The resistance to the 

 rotation is a tangential force per unit surface of the cylinder, 

 directed everywhere parallel to the motion. If the axis of 

 the cylinder be X (the electrical axis), and the radius Z, an 

 element of its surface is of the dimensions YZ. Hence, the 

 resisting stress is of the dimensions MX _1 T _2 = [E]. Thus E 

 is a tangential force per unit area. Now E has a moment 

 round X. Hence we have 



M^ 



indicating that the resisting stress over an element of surface 

 is inversely as the distance of the element from the axis of 

 rotation, and its moment directly as that distance.] 



MYZT~ 2 



8. Voltage. — Writing [E] = ^VY , we get 



™ /MYZT^X 



Now E, the line integral of E, is the source of the disturbance 

 in a closed electrical circuit. Hence, taking the above 

 mechanical illustration, E becomes the terminal torque re- 

 quired to maintain the motion of the cylinder against the 

 resistance. " The terminal torque corresponds to the im- 

 pressed voltage. It should be so distributed over the end B " 

 (in this case a rotating tube) " that the applied force there is 

 a circular tangential traction varying inversely as the dis- 

 tance from the axis" (Mr. Oliver Heaviside, Elec. Jan. 23, 

 1891). In the above case, an element of the tangent at a 

 point is Y, and of the radius Z. 



