344 



I (12) 



z =S, 



(13) 



^)l\r F ft,0' ^-^I-Vm' J 



E = (tt-) In \ dx . J £ , „cos[S e/.# .V 

 E '*,o =fe)"( n (0 ir A,] ^^o"**'*^ 



This general solution involves multiple integrals, of the 

 order 2n ; but many particular suppositions, respecting the 

 initial data, conduct to simpler expressions, among which the 

 following appear worthy of remark. 



Suppose that having assumed some particular set 

 u' l} ...u K , of values of the n arbitrary quantities u ,...u , 



we deduce a corresponding set of coefficients h\, , H^ ., by 

 the formulae (7) and (7)', and represent by s\ 2 and by 

 A 1,1' " ' A ki' '" A ni some one corresponding system of 

 quantities which satisfy the equations 



n ,2 



s (*)i a m =1 > c»r 



< A M = S (4^ )i A\ 1 ; (6r 



we shall then have, as a particular integral system, that 

 which is thus denoted : 



\,h, t = X \ A \i cos (e\ + *\ t - S (0 >\ x g .) ; (4y 

 x\ and e\ denoting here any arbitrary real quantities. If 



