ACTED UPON BY THE VIBRATIONS OF AN ELASTIC MEDIUM. 349 

 Consequently, by integrating to the second approximation, 

 c-a.e a ** = aU" y "' + aSe a x O + ***** + c x(0- 



a ™ x a 



When #, £, and -7- each = 0, 



and to the same degree of approximation k = - x'(0)- Hence c = a + 2a&. 

 Therefore, when t is not exceedingly small, 



a + 2«S - ae-™ = aiG-* (# ' S + e- x, °) - — *'(*), 

 and expanding the exponentials to terms containing m 2 , 



dx dx* 

 dl ~ lad? 



'"*{+{'- a) + XWI + **{+('- a) + X(0l)-— X(0- 

 Hence -7- = x <p (t ) , for a first approximation. 



Therefore x «) - f (« - S) (l - jMj) - # (l - 2). £| . 



to the same approximation. 



^»'W-»(«-S-m-(»-a)->('-S)-T^'-»- 



*- Bf'* --*♦(« -3*= [^F3lM 1+ S^)1] 



mrZ 1 — J" , /. x\ 

 And finally, 



dl = TTy * V ~ a) ~ «(!+*)■ + V - a) + £(T73j ' * V ' a) ' 



I shall content myself in the present communication with having 

 obtained this equation, which, as far as I am aware, is the first instance 

 of a solution of a problem of this kind. On a future occasion I propose 



Vol. VII. Past III. Qq 



