PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 21 



pendicular to OO x , and q will be ultimately the initial velocity resolved in the direction 00^. 

 Hence we have ultimately 



where, for a given direction of O x O, the integral receives the same series of values, as at increases 

 through the value 00 x , whatever be the distance of from O v Since the direction of the axis 

 of w is arbitrary, and the component of the displacement in that direction is found by multi- 

 plying by I a quantity independent of the direction of the axes, it follows that the displacement 

 itself is in the direction OO x , or in the direction of a normal to the wave. For a given direc- 

 tion of OiO, the law of disturbance is the same at one distance as at another, and the magni- 

 tude of the displacements varies inversely as at, the distance which the wave has travelled in 

 the time t. 



We get in a similar manner 



lb^^jf t jji. u o-ho) b idS, (33). 



where I, and the direction of the resolved part, g , of the initial velocity are ultimately constant, 

 and the surface of which dS is an element is ultimately plane. To find the resolved part of 

 the displacement in the direction 00,, we must suppose a; measured in that direction, and 

 therefore put I = 1, q = u , which gives £ 6 = 0. Hence the displacement now considered takes 

 place in a direction perpendicular to 00 l} or is transversal. 



For a given direction of X 0, the law of disturbance is constant, but the magnitude of the 

 displacements varies inversely as bt, the distance to which the wave has been propagated. To 

 find the displacement in any direction, OE, perpendicular to 00„ we have only to take OE 

 for the direction of the axis of w, and therefore put I = 0, and suppose u to refer to this 

 direction. 



Consider, lastly, the displacement, £ c , expressed by the last term in equation (29). The 



form of the expression shews that £ c will be a small quantity of the order — or — , since t is of 



the same order as r ; for otherwise the space T would lie outside the limits of integration, and 



the triple integral would vanish. But £ a and % b are of the order - , and therefore £ c may be 



neglected, except in the immediate neighbourhood of T. 



To see more clearly the relative magnitudes of these quantities, let v be a velocity which 

 may be used as a standard of comparison of the initial velocities, R the radius of a sphere 

 whose volume is equal to that of the space T, and compare the displacements f a , % b , £ c which 

 exist, though at different times, at the same point 0, where t ■ r. These displacements are 

 comparable with 



which are proportional to 



