PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 19 



It is to be recollected that in this and the preceding equation I denotes the cosine of the 

 angle between the axis of a> and an arbitrary radius vector drawn from 0, whose direction 

 varies from one element da of angular space to another, and that the at or bt subscribed 

 denotes that r is to be supposed equal to at or bt after differentiation. To obtain the whole 

 displacement parallel to x which exists at the end of the time t at the point O, we have only to 

 add together the second members of equations (29) and (30). The expressions for r\ and £ may 

 be written down from symmetry, or rather the axis of x may be supposed to be measured in 

 the direction in which we wish to estimate the displacement. 



20. The first of the double integrals in equations (29), (30) vanishes outside the limits of 

 the wave of dilatation, the second vanishes outside the limits of the wave of distortion. The 

 triple integrals vanish outside the outer limit of the wave of dilatation, and inside the inner 

 limit of the wave of distortion, but have finite values within the two waves and between them. 

 Hence a particle of the medium situated outside the space T does not begin to move till the 

 wave of dilatation reaches it. Its motion then commences, and does not wholly cease till 

 the wave of distortion has passed, after which the particle remains absolutely at rest. 



21. If the initial disturbance be such that there is no wave of distortion, the quantities 

 •ar', "sr", "&'", w, to", w" must be separately equal to zero, and the expression for P will be 

 reduced to £ 15 given by (25), and the expression thence derived which relates to the initial 

 displacements. The triple integral in the expression for (-, vanishes when the wave of 

 dilatation has passed, and the same is the case with the corresponding integral which depends 

 upon the initial displacements. Hence the medium returns to rest as soon as the wave of 

 dilatation has passed ; and since even in the general case each particle remains at rest until the 

 wave of dilatation reaches it, it follows that when the initial disturbance is such that no wave 

 of distortion is formed the disturbance at any time is confined to the wave of dilatation. The 

 same conclusion might have been arrived at by transforming the triple integral. 



22. When the initial motion is such that there is no wave of dilatation, as will be the 

 case when there is initially neither dilatation nor velocity of dilatation, £ will be reduced to £ 2 , 

 given by (28), and the corresponding expression involving the initial displacements. By 

 referring to the expression in Art. 17, from which the triple integral in equation (28) was 

 derived, we get 



d V rrr I d w d y d 



Now 



fff^-"^-fr-ffj('-ii?*"Si?->-'"r.?) ir - ■ ■ (31) - 

 W+ S ? dv - Iff"- S ? <"><~ff{l?} «"-fff£ ? "<>«• 



the parentheses denoting that the quantity enclosed in them is to be taken between limits. By 

 the condition of the absence of initial velocity of dilatation we have 



3—2 



