Prof. Challis on Hydrodynamics, 37 



"- a \da? + ~ay+ ~aW) " <ft 2 



__ ^r d^ d^ <P±_ 2 <i± £± 

 dx dxdt dy dydt dz dz dt 

 I have found that the equation is satisfied if 

 % = mYi sin 2q (z — /catf + c) , 

 /j being determined by the following equation, 



dr* + rdr + lbe/l -qaW ?/./' 

 The value of/ expressed in a series being known, a series for/*, 

 may be readily deduced from this equation by the method of 

 indeterminate coefficients. In this way I have obtained 



; icqi* f. * 2 + 5 g , 4*»+ll 24 "\ 



■*=— Ira 1 — 5-< + -ir^ &c -j 



There is another consideration which it will be proper to in- 

 troduce in this stage of the argument. Since the differential 

 equations which determine / and <p to quantities of the first 

 order are linear with constant coefficients, it might be supposed 

 that we may hence infer the coexistence of small vibrations. 



47T 2 « 2 



But because the value of the constant V 2 is ■ 2 (tc 2 — 1), it fol- 

 lows that the equation 



is satisfied by only a single value of X, and that for every differ- 

 ent value the equation is different. In consequence of this ana- 

 lytical circumstance, no general inference respecting the coexist- 

 ence of small vibrations can be drawn from the above equation. 

 In fact the foregoing investigations are only proper for deter- 

 mining forms of vibratory motion that are independent of arbi- 

 trary conditions. It is true that the equation might be satisfied 

 by supposing (f> to be the sum of any number of terms such as 



m sin — - (z — fcat + c), X, being the same for all; but in that case, 



as is known, the form of the sum would be the same as that of 

 each term. It may, however, be said that in virtue of this com- 

 position the factor m might be assumed to be a particular con- 

 stant, in which case the difference as to magnitude between one 

 set of vibrations and another having the same value of X would 

 depend exclusively on the number of the components and their 

 respective phases. Having premised so much as this, I proceed 

 now to show that the velocities and condensations relative to dif- 



