302 





Lord Rayleigh on the Propagation of 



respect to x, the direction of propagation, all the functions 

 being supposed proportional to e mx , where m is a complex 

 constant, and not otherwise to depend upon x. This assump- 

 tion is retained in the present paper. It seems advisable to 

 give a brief recapitulation of KirchhofFs theory, referring 

 for more detailed explanation to the original paper or to the 

 account of it in 'Theory of Sound.' 



The condition of the gas at any point x, y, z being defined 

 by the component velocities u, v, w, and #', where 6' is pro- 

 portional to the excess of temperature, the equation for 6' is 

 found to be 



WQ' - \ a 2 + K(ji! + fju" + v) \ \J 2 6 l 



+ ^{6 2 + /<^ + /,'0}V 4 ^ = 0. ..... (1) 



In this equation \/ 2 stands for d 2 /dx 2 + d 2 /dy 2 + d 2 /dz 2 ; h is 

 such that all the variables (w, v, w, 6' ) are proportional to 

 e u ; a is the velocity of sound as reckoned on Laplacean 

 principles, b the corresponding Newtonian value ; ///, fju," v 

 are coefficients of viscosity and of heat conduction. 

 A solution of (1) may be obtained in the form 



^ , =A 1 Q 1 + A 2 Q 2 , ...... (2) 



where Q lf Q 2 are functions of x, y, z satisfying respectively 



V 2 Q 1 =\ 1 Q 1 , V 2 Q 2 =^Q 2 , • • • (3> 



\i, \ 2 being the roots of 



while A l5 A 2 denote arbitrary constants. 



In correspondence with this value of 6\ particular solu- 

 tions are obtained by equating u, v, w to the differential 

 coefficients of 



BiQi + B 2 Q 2 , 



taken with respect to x, y, z. The relation of the constants 

 Bi, B 2 to A l5 A 2 is 



B^A^-i) B 2 =A 2 (v-|), 



(5> 



More general solutions may be obtained by addition to- 

 u, v, iv respectively of u' ', v', w', where u\ v\ iv* satisfy 





\J 2 v 



I 



^ 



X]' 2 i» 



-,w' 



(6) 



