IN CYLINDRICAL TUBES. 



239 



In the same manner, 



1. 



=f{aT-{%l-x)\-\ 



+ ( 



■>/^{«T-(2/+c-a!)}, 



(2 A 



&c. 



./.{a(x+?i^^^)-(2/ + c-^)}. 



&c. 



2«/ 



^{«(- + ^)-(2/-^)}. 



Hence we have at the time (t+ j, 



V =f(aT-x)-f{aT-(2l-x)} 



-{>|/[rtT-(«+c)]->|/[«T-(2/+c-a;)]} 

 — &c. 



f, , r 2(n-l)J-\ , x> . , r 2(w-l)./n -^, ,J 



+ ^|/ {« [t + — j -(21 + C-X)} 



+ &c. 



2w/^ 



+ ^{a[r + ^)-a^}-ct>{a[r+^)-i2l-a^)}, 



HH2 



