Mathematical Theory of Aerial Vibrations. 



93 



was derived from the equation (A.) in the same page, by as- 

 suming, since the motion is by hypothesis vibratory, thatjf 

 has a maximum vahie equal to unity, and that the values of jr 



1-1 ■ ^ df ^ idf ^ , 9fno« im ay>lai 

 and y which satisfy ^ =0 and ^ =0> inake , ,, . ,, „.^,, 



dx^ dip- 







On the same suppositions respecting / 1 have shown in the 

 Phil. Mag. for last December (p. ^QB) that the equation (B.) 

 is satisfied by an equation of this form, > *,-. , 



''^+«,$=o CO 



dt 



dz 



a, being a certain constant. This may be regarded as a par- 

 ticular integral applying to propagation in a single direction, 

 and is all that is required for the present investigation. An 

 integral of (B.) satisfying (1.) was also obtained (vol. xxxiii. 

 p. 363) by successive approximations. By differentiating (I.) 



and substituting the resulting values of ^-|- and -j-^. in (B.), 



we obtain 



d^^ _ 

 dz'^ " 



b''<p 





dl'' 



a^b^ 



«-G.-i 



2) 



dzdt 



--{'^-t)' 



Now substituting these values in (A.) (which equation being 

 long I dispense with inserting here), the result is 



y 



+ 2<p 



d(p 

 dz 



(-^1 



) \dx^ '^ dy^ 



) 



^ ' \dx^ dx^ doc dy dxdy dy' 



dyV 



(A'.) 



dy dxdy dy^ 

 It thus appears that the result is not independent of f un- 



