42 Rev. J. Challis on the Resistance to the Motion 



F (?•— a t) = m sin — j 1$ (a /) — f' (« /) r i 



C . Tr<p{at) Trrtt'iat) irip (at)! 

 =: VI < Sin ^^ ^—^^ COS -^—^ — -> 



I \ \ X 5 



F'(r—at) = -^-^ — - cos ^^ — — 



J ¥'(r—at) Fir— at) m . 7r<f)(a/) 



and V or — ^ — s = —r sin — -^^'^ — . 



r r- 7" X 



Now, if the motion of the disturbing surface, instead of being 

 vibratory, be continually increasing or decreasing, X must be 

 indefinitely great compared to tt <^ (a t) during the whole 



n 1 • 1 . • 7r<p(at) Tipiat) , 

 time of the motion: so that sin ^ — - = —^ — , and 



A A 



■mn^iat) u , ,, . , ^ 7r?« .^ 



V = ^— ^ — - = -^<p[at), supposing that — — = p.. At 



, . F' (r—at) irnia'iat) it , . 



the same time — ^^ = — ■ ^^—^^ — cos — - a (a ^) 



= — — J- (p' (a t), which on account of the factor r f' [a t) is 



very small compared to v. Hence in this case of disturbance, 

 the part of the velocity accompanied by change of density is 

 very small compared to the whole velocity, and therefore the 

 change of density itself is very small. 



Let, for example, ip (a t) be constant; then v varies inversely 



as r^, and ^.rf' (at) = 0, as we should expect. Again, 



let V, which we may consider to be the arbitrary velocity given 

 to the disturbing surface, be any function of the time, as/{t), 

 and let ?• = r' when t; = 0. Then r = r' + ff{t) d t, and 

 iu(p{at) = vr^ =f{t) (/ + ff{t) dty. Hence it will be 

 found that 



— —(!,'(at)= -^ ^ ^ . 



r ^ ^ a a 



2v^ 



If V be uniform, /' {t) = 0, and <p' [a t) = — 



a 



which is a quantity of an order that has been already neg- 



1 -I Tf , « I / .V ^ ^ 2 u* . 



lected. li V = gtf — — (p' (a t) = — -^ , in 



° r ^ ' a a 



