368 Proceedings of Royal Society of Edinburgh. [sess.. 
P and P\ This theorem gives explicitly and determinately the 
value of V -2 S for every point of space when 8 is known (or has 
any arbitrarily given value) for every point of space. 
§ 5. If now we put 
Hb-‘> 
we see by (5) that the complete solution of (1) is the sum of turn 
solutions , (£ 15 yj v If satisfying (6) and therefore purely distortional 
without condensation ; and (£ 2 , rj 2 , £ 2 ) w hich, in 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 iu 
either or both directions with fronts specified by (a, (3, y), the 
direction-cosines of the normal, is, with ^ and x t° denote arbitrary 
functions, 
8 =t(t- + x (t + ^ + Py± g) (10), 
where 
v — 
so that v denotes the propagational-velocity of the condensational- 
raref actional waves. By inspection without the aid of (8), we see 
that for this solution 
j 
k + ^n 
(ii); 
v 
-2 + x (t + «^±M tl!)] (12) 
For our present purpose we shall consider only waves travelling 
in one direction, and therefore take x = 0; and, for convenience 
in what follows, we shall take - f instead of v (J^j V 
f being an arbitrary function. Thus by (12) and (9) we have, for 
our condensational-rarefactional solution, 
4 _ V 2 _ £2 ax + fy + yz\ 
a 7 ; * * 
(13) 
P y 
* Camb • Phil. Trans . , November 26, 1849. Republished in vol. ii. of his 
Math • Papers, 
