330 Dr. 0. Burton on Plane and Sjrterical 



is reckoned as zero. No such simple criterion can be given 

 for the existence of a purely convergent or purely divergent 

 spherical disturbance : a fact which may be readily seen from 

 the equations for waves of infinitesimal amplitude. If $ is 

 the potential of a purely divergent system of waves, we have 



r$=f{at -r), (16) 



where / is a function whose form is unrestricted. Let p be 

 the ordinary density of the air, and p + hp the actual density 

 at a point where the coordinate is r and velocity u. We 

 have, then, on differentiating (16) the well-known relations 



_B<£_ f{at-r) f(at-r) ( 



and 



Sp__ lo<j> = f(at-r ) (18) 



p aot r ' K 



From (17) and (18), 



( a— — u\r 2 =f(at — r) , 



whence differentiating with respect to r, and neglecting small 

 quantities beyond the first order, 



i^-%)H« s f- u )- Aat - r) 



by (18) ; therefore 



op 



— ar — 



P 





«|£_ 2 " + £^_|i' =0 (19) 



p o^ v r p or - 



If, then, an infinitesimal spherical disturbance is to be purety 

 divergent, this equation must be satisfied for every value of 

 r. But since the left-hand side involves Bp/p as well as u, 

 "du/~dr, and "d (log p)/~dr, it is evident that the question whether 

 or not the equation is satisfied for some particular value of r 

 does not depend solely on the state of things in the immediate 

 neighbourhood of this value, but is influenced also by the 

 value of p corresponding to the undisturbed air. We must 

 not therefore seek to characterize a purely divergent dis- 

 turbance by a differential equation expressing that, with 

 respect to the air at each point, the disturbance is wholly 

 propagated in the positive direction of r. 



12. Not recognizing this, I had attempted to discover such 

 an equation, and one step of the inquiry is reproduced here, 

 for the sake of any interest which it may have. 



