Lord Rayleigh : Hydrodynamical Notes. 185 



Again from (2) with omission of /3, 7, 



U 2 — 2gy= const. + 2(1 —g — <x 2 — * 4 )y + a 4 cos 2a- — § a 5 cos 3#. 



.... (5) 



It appears from (5) that the surface condition can be satisfied 

 with ol only, provided that a 4 is neglected and that 



W-« a =0 (6) 



In (6) a may be replaced by a, and the equation de- 

 termines the velocity of propagation. To exhibit this we 

 must restore generality by introduction of k ( = 2-7r/A,) and 

 c the velocity of propagation, hitherto treated as unity. 

 Consideration of " dimensions " shows that (6) becomes 



h* -g-cP<*P = 9 (7) 



or 



c 2 = g/k.(l + k 2 a 2 ) (8) 



Formula* (4) and (8) are those given by Stokes in his first 

 memoir. 



By means of /3 and 7 the surface condition (2) can be 

 satisfied with inclusion of « 4 and a 5 , and from (5) we see that 

 /3 is of the order a 4 and 7 of the order a 5 . The terms to be 

 retained in (2), in addition to those given in (5), are 



2/8(1 — %J) cos 2x + 47 cos 3 x + 4a/3 cos x 



— 2/3 cos 2x + 2a/3 (cos x + cos 3x) + 47 cos 3x + 4«/3 cos x. 



Expressing the terms in cos.^ by means of y, we get 

 finally 



U 2 - 2## = const. + 2?/(l-#-a 2 -a 4 -f/3) 



-f(a 4 -f 2y8)cos2 t r-h(47~f a 5 ~2«/3)cos3^ . (9) 



In order to satisfy the surface condition of constant 

 pressure, we must take 



P=-l*, 7=A« 5 , .... (10) 



and in addition 



l-'g-cfi-iM, (11) 



correct to a 5 inclusive. The expression (1) for i/r thus assumes 

 the form 



s\r = y—ue~ y cos x -f \ afie~ 2y cos 2x — T 1 ^a 6 «"" 8y cos 3.v, . (12) 



from which y may be calculated in terms of x as far as 

 a 5 inclusive. 



