GENERAL THEORY OF SOUND WAVES 213 



which would result from initial distributions of velocity and 

 condensation expressed by 



. ............... (7) 



these functions of r being the average values of < (a, y, z) and 

 ^o (x, y, z) taken over the aforesaid sphere. It follows from 

 71 (29) that the value of <f> at P is given by 



(8) 



This gives a rule for calculating the value of < for a point P at 

 any given instant t. It may be stated in words as follows : 



To find the part of < due to the given initial distribution 

 of condensation, we describe about P a sphere of radius ct, and 

 calculate the average of the given initial values of d<f>/dt, i.e. of 

 the function ^o (#> y> z\ at the points of space through which 

 this surface passes, and multiply by t. To find the part due to 

 the initial velocities we replace the average of the given values 

 of d<f>/dt by the average of the given initial values of <, i.e. of 

 the function < (x, y, z), and differentiate the result, as thus 

 modified, with respect to t. 



The theorem contained in (8) was given by Poisson (1819); 

 the actual form (8) and the interpretation are due to Stokes 

 (1850). It will be seen that the result, as thus stated, is in 

 reality very simple, if regard be had to the great generality of 

 the circumstances which are taken into account. 



To trace the sequence of events at P we employ a series of 

 spheres whose radii (ct) increase continually from zero. If P 

 be external to the region which is the locus of the initial 

 disturbance, no effect is produced so long as the spheres do 

 not encroach on this region. If r lt r 2 be the least and greatest 

 distances of P from the boundary, the disturbance at P will 

 begin after a time r^c, will last for a time (r 2 rj/c, and will 

 then cease. 



If with the various points of the boundary of the originally 

 disturbed region as centres we describe a series of spheres of 

 radius ct, the outer sheet of the envelope of these spheres will 

 mark out the boundary of the space which has been invaded by 



\ 



