of the Motion of Fluids, Sfc. 399 



Hence, a.-'^- -^ =f\s-at- fXhAdfJ _ rPx(c>s)ds 

 So from the equations (s), 



2 dt I J a - xAc,, s)S -J a-xXc,,s) 



We are able to obtain jiarticular integrals from these equations, 

 either when P=o, or is constant. First, suppose no force to act. 

 Then, 



•> i Ja^-xic,s)S I Ja-x,(c^,s)S' 



and, because a^ hyp. log. p = - -^ ~ ~, 



2«^hyp. \o^.p=f\s-at- fJ^^^i^['-F\s^at + fxM.^)A^\ 



According to a foregoing remark, the quantities of which 

 / and F are respectively functions, are convertible into each 

 other by changing the .sign of a. Now we know, that when 

 the motions are small, a change of the sign of a has reference 

 to a change in the direction of propagation ; and that propaga- 

 tion may obtain either in a single direction, or .simultaneously 

 in oppo.site directions. Let us endeavour, therefore, to ascertain 

 whether the preceding equations will sati.sfy (9), when one of 

 the arbitrary functions, F for instance, is suppo.sed to disappear. 



In this case, 



Sato = 2a^ hyp. log. p =f{c). 



fie) 

 Hence a.=-'-^; and x(c. s) is therefore a function of c only: 



so that 



/(c) 



rx\ 



J a + 



x(c,s)ds 2a ' ttfS 



x(cr*)'"^^TM "" + '"' 

 2a 



Vol. III. Part III. 3 F 



