Manchester Memoirs, Vol. xli. (1897), No. 15. 27 



The t outside may be replaced by r/b, and the fraction 

 rj{r-\-r'), being sensibly constant over the range of inte- 

 gration, may be put outside. Our expression then becomes 



4^^//(«o-#)^.* 



As the disturbance deemed initial was only a momentary 

 condition of a wave that had been travelling outwards 



with the velocity b, we must have u 0 = — bjp, and 



therefore 



1 rr 



The expression is left in the first instance in this 

 shape in order to show more clearly the manner in 

 which each portion of the disturbance in the state 

 taken as initial contributes towards the future distur- 

 bance at P. When there is no obstacle to the transmis- 

 sion we shall have J d<p=2ir, and J" {^) dr'—(£o) taken 



between limits. If bt <P Q, the sphere round P with 

 radius b t does not cut the disturbed region at all, 

 and the disturbance at P is nil. If bt>PS, the 

 limits of r' are the distances from 0 at which the 

 sphere round P cuts the inner and outer limits of the 

 shell, and as the disturbance there vanishes, we have 

 again no disturbance at P. But if bt lies between 

 those limits, and the sphere round P cuts 0 P in T 

 (which point must lie between Q and S) the limits of 

 r'^will be 0 T to a point in the outer boundary of the 

 shell, where therefore ^ 0 vanishes. Hence the displace- 

 ment at P is the same as was initially at T, only 

 diminished in the ratio of r-\-r r to r' , as we know it 

 ought to be. 



* The suffix b t means that the integration is taken over a spherical 

 surface with centre P and radius bt. 



