Prof. Challis on the Principles of Hydrodynamics. 91 



making pro hac vice the hypothesis that the increment of elasti- 

 city varies at each instant from point to point, and is exactly 

 proportional to the increment of density. A theory of this kind, 

 resting on hypotheses, ought to have no weight against a course 

 of reasoning which deduces the velocity of sound exclusively 

 from hydrodynamical principles. 



From what has now been proved, it appears that the velocity 

 (V), so far as it is independent of any arbitrary disturbance, may 

 for small motions be expressed by the function 



m/sin-r- {z + Kat'\-c)y 



A/ 



the quantities m, X, and c being altogether arbitrary. The 

 numerical quantity k is equal to 1-185447. Also from the 

 equation 



it follows that the condensation {a) is at the same time expressed 



by the function 



, Kmf . 2iT , _ ^ . . 

 4- — -mn—-{s-v-Kat + c). 



Hence between V and cr we have the general relation 



the -f or — sign applying according as the propagation is in 

 the positive or negative direction. 



The application of the foregoing general results to particular 

 cases of disturbance is to be made on the principle, that the 

 initial disturbance and subsequent motion must be composed of 

 parts, either finite, or indefinitely small, that conform to the 

 circumstances of the motion that have been shown to be inde- 

 pendent of all that is arbitrary. There are two distinct classes 

 of disturbance of which a compressible fluid appears to be sus- 

 ceptible. In the one class no velocity is impressed initially, and 

 the condensation through a space of arbitrary extent is made to 

 be of arbitrary value at a given instant, after which the fluid is 

 allowed to move freely. It may be conceived that the initial 

 state results from a combination of the motions obtained in the 

 foregoing investigation, which for distinction 1 shall call normal 

 motions, the number of the sets of vibrations, the directions of 

 their axes, and the values of the quantities m, X, and c being 

 completely at our disposal. The fluid being left to move freely 

 after the first instant, it may be presumed that the values of m, 

 X and c, which belong initially to a given set of vibrations, remain 

 unaltered. Although the value of the function / indicates that 

 the transverse motion in a single set of vibrations extends inde* 



H2 



