252 
being that the velocities v, of all these particles have the same value 
but different directions which succeed each other at equal angular 
intervals. The only component that contributes a part to the scalar 
product, is vo, which always has the same direction as v,. As the 
component of the angular velocity about the axis through WW paral- 
lel to ON is equal to— y sind, we find for m(v, . v,) 
— mrp. rp sin 6. 
Taking the sum for all the circulating electrons, we find 
— Wy sind. 3E mr’? = — D wy sin @. 
Thus the whole kinetic energy becomes 
Dee tA p° sin O+ BO + Cg’ cos? 6 + D we -2Dw 7 sin A} 
Writing A+ D for A, we get 
Beep {A g?sint A+ BO LC pr cos? 0 + D(p — y sin A)°} 
So that the force tending to increase w is given by 
ror oT d 
y= - =) SS 
dt \ Ow oY dt 
We shall start from the supposition that there are no forces acting 
on the electrons, by which their velocities in their paths might be 
changed. We then have 
D (w—y sin a), ; 
Win OF W—y dn. B nst et ie sO) 
and we find instead of (2) 
19 ; : 
Q—B is +(C — A)g?sn@cos6 + Dy yeosd . . (da) 
a 
We have seen already, that (’ must be greater than A. 
79 
a . 
In the stationary state, in which @ does not change, on will be 
a 
equal to zero; so that in the absence of a couple 9, the angle 4 will 
assume a constant value given by 
DEN 
CAS 
The same formula may be found in a somewhat simpler way 
by considering the moments of momentum. We then must apply 
the following principle. In any system the change of the resultant 
moment of momentum has the direction of the moment of the 
couple that gives rise to this change. 
In our case the only couple acting on the system arises from the 
forces applied at the extremities of the axis OY, by which this axis 
is compelled to move in a horizontal plane with the constant angular 
velocity ¢, The axis of this couple lies in the plane YOZ. Indepen- 
UN 
