1921-22.] The Faraday-Tube Theory of Electro-Magnetism. 245 
Translational motion of the body will not affect the space relations 
of jj., but rotation will. Any vector a in the body will in virtue of rotation 
w change at the rate 
a = Vwa , 
and, as is well known in quaternions, the rate of change of jua- will be 
= V (jO/XCT — /xV cocr + /X(T . 
Hence the equation of linear motion is 
/3 = Vco/xcr — /xVcocr + ftor . . , . . (1) 
The moment of momentum of the body with regard to a fixed origin 
is of the form 
y = V/0/XO-+ <j5>(o , 
where p is the vector position of a given point of the body and (p is the 
linear vector function which, operating on the angular velocity gives the 
angular momentum about the extremity of p, (See Tait’s Quaternions, 
p. 323.) 
If ijj is the torque acting on the body, the equation of rotational 
motion is 
= y pfxcr p _ (pLcr) poi + <p(h 
Ctt 
or 
= Y o-/xcr + V pf 3 + y (0^0) + pih 
i/^ = y(r/xcr+</Kb + yaHj6w ...... (2) 
The total activity equation is obtained from (1) and (2) in the form 
Serfs + Sco^ = ^{^SerpLor + 
the integral of which may be represented in the form 
W = T-T,, 
where W is the work done by the forces /3 and ip, and T( = T 3 ^H-T 2 ) is the 
kinetic energy. It will be seen that T is a quadratic function of the 
components of the velocity and of the angular velocity, the two parts 
T^ and Tg being quite separate, and no products of the components of 
different type existing. This feature involves or is involved in the self- 
conjugateness of the operators. 
If cr is parallel to one of the axes of pb, so also is p.ar, and YarjULo- vanishes. 
Equation (2) then reduces to 
ij/ = (poi + y , 
