504 Mr. Tovey’s Researches in the 
If we conceive a slowly tapering cone (fig. 1.) to have its 
Fig. 1. B E 
A A sill eae aee pe eebveed 
C D 
summit A at the centre of agitation of a system of spherical 
waves, and if we take the axis of the cone for the axis of 2, 
it is clear that the displacements £, 4, % of the molecules 
within the frustum B C D E may be regarded as functions of 
x and ¢; and may therefore be expressed by the equations 
(2.), nearly. It is also manifest that the same equations will 
express the displacements for any other frustum of the me- 
dium, by making the arbitrary quantities to vary according to 
the position of the frustum. Consequently, if we suppose 
= asin(nt—kx+a) 
for the frustum B C D E, the same equation may be taken to 
express the value of & for any other frustum of the same cone, 
by regarding a, 2, k, a as functions of x. 
Let g be the radius of the sphere of influence of the mole- 
cules: then, if = were infinitely small, the minute portion of 
a wave contained within the sphere would be a plane wave, 
and a, 2, k,a constant. Hence we perceive that these quan- 
tities must be functions of a and consequently, that we may 
write 
2 
ee Ay B £+4c(4) + &e., 
x x 
niles, Ata +0 (2) + &c., 
x x 
g 
x 
2 
gee Am 4 BM 2 4 Om/ g ) aoa 
x 
the only variable quantity in these series being z. 
Now when z is infinite « must be zero; therefore A = 0: 
and as 8. is, at all sensible distances from the centre of agi- 
