ACTED UPON BY THE VIBRATIONS OF AN ELASTIC MEDIUM. 341 



an equation in which u and / are the only variables, and which serves 

 to determine the value of the function f{r — at) from the given value 

 of <j>(t). This equation gives by integration, 



_at _at at 



u = Ce r — mare r fe r <p(t)dt, 



du Ca — — — — — 



-r. = e r - mare r fe r <p'(t)dt. 



Now -j- = - a f'(r - at) = r-£. Therefore 



deb Ca -% -^ r- , 



dt = ~~r^ e "- mae r fe r <p\t)dt (8), 



As an example of the application of this formula, let us suppose 

 as before that <p{t) = sin ^_* . Then 0'(/) = *^cos— '. Also 



e cos — - — dt = 



T 2tt«/ _, \r XX X 



7 (a 2irat Zira . 2nat\ 



eM-.cos — — — — sin 



r* + \* 



at 



\e r . (2-n-at 



.sin i 



2ira 



(%-n-at \ 

 a COS (_ - «) , 



by substituting tan a for . Hence 



rf0 Ca -g . /27ra# 



= «■ e — 7»a sin i 



a*rf 



r 



.2 



a COS ( — a 1 . 



and consequently by equation (6), 

 «<Nap. log.p = Ca e -1 + Hasina cos (!=2f - a) - |%in*^ (9). 



It will be seen by the above result that the term of equation (7), 

 retained in this instance, introduces into the expression for p a quantity 



of the order of — x sin 2 a, or of - x r?, and therefore of the third order, 

 fit o, \ 



Hence that term is more considerable than the other small terms of 



equation (7), and we may be confident that by retaining it and rejecting 



Vol. VII. Part III. P P 



