and Refraction of Solitary Plane Waves. 181 



virtue of (9) , is irrotational and involves essentially rarefaction 

 or condensation or both. This most important and interesting 

 theorem is, I believe, originally due to Stokes. It certainly 

 was given for the first time explicitly and clearly in § § 5-8 

 of his " Dynamical Theory of Diffraction '"'*. 



§ 6. The complete solution of (3) for plane waves travelling 

 in either or both directions with fronts specified by (a, ft 7), 

 the direction-cosines of the normal, is, with ty and % to denote 

 arbitrary functions, 



s = ^(,_^±to) +x (, + f£±to) (10) , 



where 



e=?v /*+t? (11) . 



so that v denotes the propagational-velocity of the con- 

 den sational-rarefactional waves. By inspection without the 

 aid of (8), we see that for this solution 



v^-K!H*H**^ + x(. + =**±*!)i] 



For our present purpose we shall consider only waves 

 travelling in one direction, and therefore take %=0; and, 



(d \ — * 

 T.J f 



instead of v( -rj ^ ;f being an arbitrary function. Thus 



by (12) and (9) we have, for our condensational-rarefactional 

 solution, 



k_ 7 h__k_, f( t gg±gy±3? \ . . (13). 

 * ft y J \ V J 



In the wave-system thus expressed the motion of each 

 particle of the medium is perpendicular to the wave- front 

 (a, ft 7). For purely distortional motion, and wave-front 

 still (a, ft 7) and therefore motion of the medium everywhere 

 perpendicular to (a, ft 7), or in the wave-front, we find 

 similarly from (7) and (6) 



fi - vi _ Sl rls ax+ fy+v z \ (u), 



^A" i 8B-7C" / V u / ' 



where 



••■V7 (15) ' 



* Camb. Phil. Trans., Nov. 26, 1849. Republished in vol. ii. of his 

 ' Mathematical Papers.' 



Phil. Mag. S. 5. Vol. 47. No. 285. Feb. 1899. 



