580] Equation of Wave-propagation 505 



which will be denoted by S. Let us first calculate the value of the contribu- 

 tion to equation (544) from the first surface. We have, on this first surface, 



dn dr r 2 r 



so that when r is made to vanish in the limit, we have 



rr/^8^_^a 



and therefore 



4-7T 



since the integrand vanishes except when t = 0. 

 Thus equation (544) becomes 



_ r 



(545). 



T 



Integrating by parts, we have, as the value of the first term under the 

 time integral, 



't" ^ 9,, 

 -t r r dn 



t=t" rt" j fl r dQ 



T f r _j_ at\ I 



t=-t' J -t' ar d n dt 



The first term vanishes at both limits, and equation (545) now becomes 



3V=o = T^ M dS I F(r + at)\ - -j- <l> ^- f - ] + - -~-[ dt. 

 t=o 4?r JJ J -f (ar dn dt dn \rj r dn) 



We can now integrate with respect to the time, for F (r + at) exists only 



at the instant t = . Thus the equation becomes 



a 



M*f| -- -3>-(i)+i r dS, 



giving the value of $> at the time t = in terms of the values of 4> and <t> 

 taken at previous instants over any surface surrounding the point. The 



