22 PROFESSOR STOKES, ON THE DYNAMICAL THEORY OF DIFFRACTION. 

 But, in order that the triple integral in (29) may not wholly vanish, - must lie between the 



T 



limits - and - , or at most lie a very little outside these limits, which it may do in conse- 

 nt 6 



quence of the finite thickness of the two waves. Hence the quantity neglected in neglecting £ 



R 



is of the order — compared with the quantities retained. 

 r 



The important terms in the disturbance due to the initial displacements might be got from 

 equation (30), but they may be deduced immediately from the corresponding terms in the dis- 

 turbance due to the initial velocities by the theorem of Art. 14. 



24. If we confine our attention to the terms which vary ultimately inversely as the 

 distance, and which alone are sensible at a great distance from T, we shall be able, by means 

 of the formulae of the preceding article, to obtain a clear conception of the motion which takes 

 place, and of its connexion with the initial disturbance. 



From the fixed point 0„ draw in any direction the right line 0,0 equal to r, r being so 

 large that the angle subtended at by any two elements of T is very small ; and let it be 

 required to consider the disturbance at 0. Draw a plane P perpendicular to 00 a and cutting 

 00i produced at a distance p from 0,. Let - p u + p 2 be the two extreme values of p for 

 which the plane P cuts the space T. Conceive the displacements and velocities resolved in 

 three rectangular directions, the first of these, to which £ and u relate, being the direction 00,. 

 Let f u (p>, f v (|>), f a (p) be three functions of p defined by the equations 



/. (P) = jfadS, /, (p) = ffv dS, /. (p) - ffw dS, . . (34) 

 and /j (p), f v (p), / f (p) three other functions depending on the initial displacements as the 

 first three do on the initial velocities, so that 



ft GO = fftodS, /„ (J>) = //„, dS, ft (p) = //£, dS. . . . (35). 



These functions, it will be observed, vanish when the variable lies outside of the limits - p, 

 and + p 2 . They depend upon the direction 0,0, so that in passing to another direction their 

 values change, as well as the limits of the variable between which they differ from zero. It 

 may be remarked however that in passing from any one direction to its opposite the functions 

 receive the same values, as the variable decreases from + pi to — p 2 , that they before received 

 as the variable increased from — p, to + p 2 , provided the directions in which the displacements 

 are resolved, as well as the sides towards which the resolved parts are reckoned positive, are 

 the same in the two cases. 



The medium about remains at rest until the end of the time a~ l (r-p l ), when the 

 wave of dilatation reaches 0. During the passage of this wave, the displacements and velocities 

 are given by the equations 



4 irar %irr 



« - T— A' (at -r) + ~f $ "(at -r), v = 0, w = 0. 



47TP %7TT 



(36). 



