FROM A VIBRATING BODY TO A SURROUNDING GAS. 
453 
the sphere is such that the normal velocity of its surface is expressed by a Laplace’s 
Function of any one order, the disturbance at a great distance from the sphere will vary 
from one direction to another according to the same law as if lateral motion had been 
prevented, the amplitude of excursion at a given distance from the centre varying in 
both cases as the amplitude of excursion, in a normal direction, of the surface of the 
sphere itself. The only difference is that expressed by the symbolic ratio F n (c ) : F 0 (c). 
If we suppose F n {c) reduced to the form [A n (cos u n +■ \/ — 1 sin a n ), the amplitude of 
vibration in the actual case will be to that in the supposed case as [j, 0 to (*„, and the 
phases in the two cases will differ by a 0 — a n . 
If the normal velocity of the surface of the sphere be not expressible by a single 
Laplace’s Function, but only by a series, finite or infinite, of such functions, the dis- 
turbance at a given great distance from the centre will no longer vary from one direction 
to another according to the same law as the normal velocity of the surface of the sphere, 
since the modulus [h n and likewise the amplitude a n of the imaginary quantity F„(c) vary 
with the order of the function. 
Let us now suppose the disturbance expressed by a Laplace’s Function of some one 
order, and seek the numerical value of the alteration of intensity at a distance, produced 
by the lateral motion which actually exists. 
The intensity will be measured by the vis viva produced in a given time, and conse- 
quently will vary as the density multiplied by the velocity of propagation multiplied by 
the square of the amplitude of vibration. It is the last factor alone that is different 
from what it would have been if there had been no lateral motion. The amplitude is 
altered in the proportion of (Jj 0 to so that if 
■l n is the quantity by which the intensity which would have existed if the fluid had been 
hindered from lateral motion has to be divided. 
For the first five orders the values of the function F„(c) are as follows : — 
F 0 (c)=mc-j- 
Fj {c)—imc-\- 
F 2 (c)=«mc-f~ 4 +t^--}- ,. L 
v ' 1 imc 1 ( imcy 
F 3 (c)=ime+ 7+j£+^+^» 
t, , , . -... 65 , 240 , 525 , 525 
F 4 (tf)_*mc+ll+j— H-^5+^p+(^4- 
If A be the length of the sound-wave corresponding to the period of the vibration, 
m=™, so that me is the ratio of the circumference of the sphere to the length of a 
