ACTED UPON BY THE VIBRATIONS OF AN ELASTIC MEDIUM. 339 



Again, let us retain the term —■-?-■ , ,.. , rejecting the others. Then 



d*<j> _ 2 dj> d 2 <f> 2dcj> 



dr 11 a 2 ' dr ' drdt rdr ~ ' 



dv 2 d 2 (p 2 

 tar a 2 ' drdt r 



Hence, integrating with respect to r, 



2 d<t> 

 Nap. log. vr 2 - - 2 . -^ = Nap. log./(0; 



.-. vr 2 = f(t) .&% =f(t) • (l + |. -£) , nearly. 

 Under the same conditions as in the last case, the first approximations 



to the value of v and -3? are ^-^ and - — — . Substituting- this latter 



at r* r & 



quantity in the above equation, 



dr r* «y ' 



and integrating with respect to r, without adding an arbitrary function 

 of the time, 



d,= /(*> . /CO/'**) . 



v r d'r* ' 



.-. ** = _•££> . l /'(0}'+/(0/'(0 

 flfa r aV- 



To take a particular instance, let the velocity impressed at the time t 

 at the distance r from the centre, and in the direction of this radius, 



be m sin — - — , a being supposed very large compared to m, and X very 



large compared to r. Then, for first approximations, 



, Vj ,. , . 2Trat „,.. 2-namr 2 9,-n-at , „,,.. 4nrWmr- . 2ttuI 

 f (t) = m r 2 sin -y- ,f(t) = — — - cos — , and/ (0 - sin -^— 



These values will enable us to estimate the order of the second term of 

 the above expression for -j- . By substitution they give for this term, 



