Prof. Challis on some Theorems in Hydrodynamics, 341 



pressions for the velocity and density. It will also appear from 

 considerations which follow, that no other general supposition 

 respecting propagation is admissible. 



If p = \ +0; and the product Vo- be neglected, the equations 

 become 



Having thus far obtained results confirmatory of those of 

 Proposition XII., I proceed next to treat in a new manner Pro- 

 position XIII., which relates to the motion and velocity of pro- 

 pagation in a rectilinear tube of arbitrary and indefinitely small 

 transverse section. 



The investigation of the laws of the mutual action of the parts 

 of a compressible fluid in Proposition X. (Philosophical Magazine 

 for December 1852), led by a general process to the conclusion 

 that the motion takes place along a rectilinear axis, and that the 

 rate of propagation of the velocity and density along the axis is 

 constant. On the suppositions that the propagated motion is 

 small, and that no impressed force acts, particular functions 

 expressing the velocity of the fluid parallel and transverse to the 

 axis and the density of the fluid were obtained. Also (Prop. XI.) 

 a numerical value of the rate of propagation in air was found 

 without reference to any arbitrary disturbance. The above- 

 mentioned functions appear to apply to cases in which the motion 

 results from the mutual action of the parts of the fluid, such, for 

 instance, as might be caused by the confluence of two streams, 

 or by the passage of a uniform stream over the mouth of a tube. 

 In my communication to the Philosophical Magazine for last 

 February, I have called the motions defined by these functions, 

 normal motions. The term was not happily chosen, and being 

 liable to be misunderstood, I propose to substitute for it free 

 motions. The course of the inquiry now leads us to deduce from 

 the laws of free motion, equations applicable to motion which is 

 constrained to take place in straight slender tubes. For the 

 sake of simplicity, the reasoning will be confined to small quan- 

 tities of the first order of approximation with respect to the velo- 

 city and condensation. 



The motions parallel and transverse to an axis of free motion 

 are defined generally by the two equations, 



<£=//, cos — (#— icat-\-c), 



f=l-e(x* + y% 

 in the immediate neighbourhood of the axis, Hence, as the 



