362 Prof. Challis's Determination of the Velocity of Sound 



«=«'nS» + #,) +*y^ T7 5A^ 



-^t{--4M-¥) 



r • (A'.) 



^ \dx^ ' dx^^ da;' dy' dxdy dy"^ ' dy^) ' 



Hence it appears that an equation altogether independent of 

 f is obtained by neglecting powers of this quantity above the 

 first. When this is done, the equation becomes 



d^f dH by ,^. 



At the same time equation (B.) becomes 



-§-'S-*^^ • • ; ■ (-) 



We have thus arrived at two equations, one of which shows 

 thatyis a function of ^ and y only, and the other that <p is a 

 function of ;s and t only. These results are in accordance 

 with the original suppositions respecting these quantities, by 

 which /^c/o^ + uc^^z + wc^z was made integrable. The condition 

 of integrability of that quantity has therefore now been satis- 

 fied in a manner consistent with the hydrodynamical equations. 

 It is to be remarked that the equations (^.) and (y.) contain 

 only the first powers of / and f. Now since the reasoning 

 given at the commencement of this communication, by which 

 It was shown that udx + vdy + wdz must be Integrable for vi- 

 bratory motion, extended only to the first power of the velo- 

 city, there was no reason to expect greater generality in the 

 equations which determine f and f. It may, however, be 

 remarked, that the equation (/3.), as equation (A'.) shows, is 

 satisfied without any restriction of the value of <p, the more 

 exactly in proportion as the value of/" approaches more nearly 

 to unity. We may hence conclude that for points on and imme- 

 diately contiguous to the axis of %, the motion is exactly de- 

 termined by the equations (/3.) and (B.), and that for these 

 points udx + vdy + wdz is integrable for exact values of u, v, 

 and TO. This result is important, because it enables us to infer 

 from the reasoning that has preceded, that along the axis of 

 z waves are propagated without undergoing any change what- 

 ever, all the parts of a given wave being propagated with pre- 

 cisely the same velocity. 



To obtain a complete idea of the nature of the motion, it is 



