74 Prof. E. H. Barton on Spherical 
Then it is shown in text-books on the calculus (e. g. William- 
son's ' Differential Calculus ') that 
2 tPs 2ds , 1 d( . n ds\ , 1 d 2 s .. _ 
c/r- ?* rt?> r- sin vdd\ ad' r 2 sm 2 d<j) 2 v y 
Now suppose we have symmetry about the centre of the 
system, so that s is a function of r only. Then ds/d6 = and 
dsld(b = 0. Hence, for this case, (15) reduces to 
, d 2 s 2 ds 1 d 2 (rs) . lfi * 
rt/ v - r rtr r dr 2 
Equation (11) accordingly becomes 
**&-*& (17) 
The general solution of this may be written 
rs=f &-<*)+/ Jr + a£) (18) 
where f\ and / 2 denote arbitrary functions. This solution 
obviously consists of both diverging and converging waves 
of spherical form, of any periodic or non-periodic type and 
travelling with radial speed a. For diverging waves of 
simple harmonic type (18) becomes 
s= - cos k(r— at) (19) 
To denote the speed u of the air along the radius r, we 
derive from (6) and (19) 
Thus 
a 
= — a 2 1 ydt = — \ cos k(r— at)— -=- sin k(r— at) 
J dr r \ v fcr y J 
(21) 
Again, if displacements along r be denoted by f, we have 
by another integration 
£= i w^/, or 
?=-£{^*('-<>+^os*<V-aO}. • ( 22 ) 
"We may thus see, from equations (19), (21), and (22), that 
it is only the condensation s whose change of phase is 
restricted to the ordinary one inseparably associated with the 
