18G6.] 



and on Partial Differential Equations. 



307 



For La Grange's equations we may substitute the following, viz. — 



(A) 



dv , ai^ dp ^ 



di~^l^dx~ ' 



dv _^J) dp Q 



dx f dt 



The first of these is obtained from the equation of the Encyc. Met. 



(Art. " Sound"), by putting v for ^ and ? for 



dt p dx 



The second results from a similar substitution in the analytical condition, 

 d_(dy\^d^rdy\ 

 dx\dt) dt\dx)* 



Poisson's solution has always been regarded as imperfect, and may 

 easily be shown to be so. 



I was some time ago struck by finding that the above equations, while 

 they yielded with facility the result of Poisson, notwithstanding their 

 simplicity, baffled every effort to extract from them one more consonant to 

 the general exigencies of the problem. 



I had at this time arrived at the conclusion that the law of pressure 

 assumed in the received theory of Fluid Motion could not be generally 

 true, and in a Paper* communicated to the Royal Society, had pointed 

 out that, in a certain case of motion, the assumption of the truth of that 

 law led to a contradiction ; while in another case of motion the expression 

 for the pressure given by the received theory was palpably erroneous. 



It occurred to me, therefore, that the defective law of pressure of the 

 received theory accounted for the defective solution which alone was ob- 

 tainable from the equations of motion derived from it. If the law of pres- 

 sure of the received theory was not always true, if it held only when cer- 

 tain conditions were satisfied, those conditions would obviously have the 

 effect of dismissing from the complete solution of the problem obtained on 

 a perfect theory at least one of the two arbitrary functions which it must 

 necessarily involve. 



With a view to establishing this point, assume the solution of the equa- 

 tions of motion to be contained in the pair of equations, 

 '^{xytpv) = ^{f{xytpv) }, | 

 Yixytpv):=^l^{f{xytpv)},\ ' 



If these equations satisfy the equations (A), the latter will equally be 

 satisfied by the pair of equations, 

 Y{xytpv^=v, 

 F(xijfpv) = xb{f{xyfpv)}. 



But it is shown in my Paper that these latter can only satisfy the equa- 

 tions (A) on the supposition that F is of the form 

 F=F(y±«log,p). 



jNIoreover, since we have in equations (B) the function F equal to an 



^- On the True Theory of Pressure as applied to Elastic Fluids in Motion. 



2 c 2 



