234 Proceedings of the Royal Society of Edinburgh. [Sess. 
equation (35) above^ the stress to which the restoring force is due will 
now be the quasi-tension E + VqB, where q is the velocity of the tubes, of 
which we shall suppose that only one set need be taken into account ; and 
with this last assumption we ma}^' drop the suffix m and so write 
B = /^VqD, E = |. 
The d component of E + VqD is the only effective part of the stress, and 
its magnitude is given by 
(E + YqB)d, = f i + yuVqVqdy , 
where is the unit vector parallel to d, or d = dd^. This equals 
g(l+;«KdiVqVqd) 
= |{l-MK(Vd,q)2} 
where cVK = l. 
Kl c2j ’ 
(41) 
The linear density will remain ij.d, so that the velocity of propagation 
along the tube will be — Since the tube itself is in motion 
with velocity 'i; in a perpendicular direction, the propagation of the 
disturbance in space will be with velocity c in a direction making an 
angle sin “ ’ — with the tube. When v = c the disturbance will not be pro- 
pagated at all along the tube, which will lie in the wave-front ; and the 
traction (E + /xVqVqD) will vanish. 
9. To take into account a general velocity of the tube in the direction 
of its length, let us restrict ourselves to plane-polarised radiation. We 
shall take the a?-axis in the direction of propagation, and the ^-axis in that 
of the disturbance. Since we are dealing only with transverse vibrations, 
the velocity of the tubes in the direction of the ray will be constant from 
point to point along a tube. Let be this a?-component of velocity. Also 
let {x, y) be the coordinates of a point on some particular tube at time t, 
so that y is a function of x and t. Then the whole ^/-component of velocity 
of the point will be 
dy di/ dy 
-y = + u— 
dt dt~ dx 
. (42) 
It is obvious that the shearing motion perpendicular to the a?-axis of 
the tubes in their vibration will not affect the number of tubes per unit 
area passing through a plane normal to the ^r-axis. Thus the quantity 
