Manchester Memoirs, Vol. xlii. (1898), No. 9. 15 



above. The solutions of (58) and (59) which are finite 

 for r = oo may be written 



where A, B are arbitrary constants. Substituting in (55) 

 and (56), and eliminating the ratio AjB^ we find 



This equation, in conjunction with (18) and (57), deter- 

 mines the wave-velocity c. 



In the present investigation, rja and Za are real, and 

 less than 7na, which is by hypothesis small. Now we 

 have seen from (39) that for small values of ^ the fraction 



is of the order log ^. The terms in (61) which have a 

 coefficient of this form are therefore of the order ^^ log ^ 

 as compared with the remainder, and may (for sufficiently 

 long waves) be neglected. The equation then takes the 

 simple form 



^v''-lvfi = g/ = y:c'\c^ . . . (62), 



where k denotes the cubical elasticity of the fluid. If we 

 substitute the value of v^ from (18), this gives 



^r^^i^ ■ • •, ■ <^3)- 



The value of ^ for glass is (roughly speaking) about 10 

 times the value of k for water. This would give a 

 diminution of about 5 per cent, in the velocity of sound in 

 the water. 



The formula (6^) is precisely what we should obtain 



