Analytical Considerations respecting the Velocity of Sound. 99 



respectively by quantities of a higher order than the first. As 

 I am about to treat of small vibratory motion by making use 

 of exact equations, I shall assume, in accordance with the 

 principle stated by Mr. Stokes, that udx + vdy + isodz is accu- 

 rately an exact differential. Let, therefore, udx + vdy + wdz 

 = (d.<pf) i <p being a function of z and t only, and/a function 

 of x and y only. This being premised, we may proceed to 

 employ the exact equation (n) obtained by Lagrange in the 

 Mecanique Analytique (part 2. section 12. art. 8). By sub- 

 stituting <p/for <p that equation becomes, 



>-<•{*(§ 



+ 



+/ 



dy 2 ) VJ dz 2 j J dt 2 



-f- 



9 dt \dx 2 + dy 2 ) lJ dz dzdt 



7 \dx 2 ' dx 2 "I" dy 2 ' dy 2 ) J dz 2 ' dz 2 

 ,¥dj_^d?tdp dp 



h (A.) 



•2<p 



w- 



+ 



dx dy dxdy ~ rj dz 2 \dx 2 ' dy 2 



The motion being by hypothesis vibratory, it may be as- 

 sumed that the function/ has a maximum value, which, since/ 

 is numerical, may be taken for the unit of value. Let there- 



fore the values of x and y given by the equations -j- = and 



-r-=0 make/=l, and at the same time make 



dx 



<Pf d 2 f_ _b 2 

 f df ~ d» 



dx 2 



the negative sign being a consequence of the supposition of a 

 maximum. On account of the independence of the variables 

 in y and <p, we obtain by substituting these values in the equa- 

 tion (A.), the following equation for determining <p: — 



<P<p d 2 <p dpd*<p _dtf d?<p 



[IS. J 



+ a <>~* l dzdzdt dz 2 



dz 2 dt 2 



d 2 ? 

 dz 2 ' 



Again, to obtain the equation which determines / let us 

 suppose, on account of the independence of the variables in 



0; and let the value of z given by this equa- 



f and/, that ■£ 



tion be substituted in <p, which will thus become <p , a function 

 of t or a constant. Now substituting these values in (A.), and 

 observing that equation (B.) gives 



H 2 



0: 



df 



