Mr. 11. H. M. Bosanquct on the llieory of So mid. o5 



^ this t 

 the rate of 



Now this term signifies a transmission of energy outwards at 



-- —J per second per unit surface ; 



that is to say^ the energy through surface Ttr^ gradually dimi- 

 nishes in the ratio of -^ : it disappears ; there is nowhere for 



it to go to. It is clear, therefore, that this is not a solution as far 

 as the flow of energy is concerned. 



It is onh^ necessary to take up integral (ii), and out of (i) 

 and (ii) to combine a solution which shall be consistent with 

 the energy-conditions. Of course the origin of r is so far 

 indeterminate, as well as the origin of time. We here intro- 

 duce a constant c ; and the following is the expression for the 

 resulting velocity, deduced as a combination of integrals (i) 

 and (ii) of the differential equation of symmetrical spherical 

 divergence : — 



. . Y S . 

 velocities ^^W^ sin/c(v^— r + c) divergent, 



V S 



-^^ sin hivt^-r — c) convergent, 



which combined give the velocity of a region in the neigh- 

 bourhood of a loop surface, 



V S -. 

 ^ ^ sin kvt cos k{r — c) , 



or, since k{r — c) is small, 



V s 



— |r-^sin^vi; 



b 



that is to say, the differential equation is satisfied by the mo- 

 tion of a stationary wave near its loop (therefore unaccom- 

 panied by changes of density), and the energy-conditions 

 are satisfied by representing it as the sum of a divergent and 

 a convergent stream. 



It remains to show how either of the component streams 

 may be regarded as made up of the elements of energy reflected 

 from the other at the different surfaces S. It is easy to see 

 that the result of such a number of reflexions may be repre- 

 sented by supposing them collected into one reflexion at a dis- 

 tance T from the centre. In the expressions for the divergent 

 andconvergent velocities put c = r; then the position denoted 

 by r will correspond to the actual loop surface of the stationary 



D2 



