94 Prof. Challis on the Principles of Hydrodynamics. 

 to the first approximation we have 



V<fe "^ R "^ R' " ydt' . 



Supposing this equation to apply to constrained uniform recti- 

 linear propagation, we have dis=fl?R=«?R'; and from the equa- 

 tions above, omitting the arbitrary quantity ^^(w, t), 



da _ <f>\s^a!t-\-c) 

 "Ydt" <l>{s—a't + c) ' 

 Hence by integration 



v=±W 



Since by comparison with the foregoing equations S4r does not 



contain the time, we must either have RR' an arbitrary function 

 of s and yjr^t) a constant, which is the case of motion in a rigid 

 tube, or simply '\jr{t) a constant, which is the case of motion 

 constrained to be rectilinear, and at given positions to be in the 

 directions of the normals to a given surface, but in other respects 

 free. For uniform propagation from a centre R = R', and 



V=a'(7=^,</>(R-a7 + c). 



Proposition XIII. To prove that the propagation of motion 

 in a rectilinear tube of arbitrary and indefinitely small trans- 

 verse section is uniform^ and that the rate of propagation is the 

 same as that along a rectilinear axis of free motion. 



In the first place it is to be remarked, that the condensation 

 and velocity may vary in a manner entirely arbitrary, both at a 

 given position in successive instants, and from point to point of 

 the axis of the tube at a given instant. This follows from the 

 fundamental principle that the parts of the fluid may be momen- 

 tarily separated by an indefinitely thin partition transverse to the 

 motion, provided the condensations and velocities on the oppo- 

 site sides of the partition are equal. Assuming, therefore, the 

 motion to be made up of parts of the normal motions, the quan- 

 tities m, \ c cannot generally be considered constant except 

 through an indefinitely small space, or for an indefinitely small 

 time. In fact, the general motion becomes identical with the 

 particular case, by making these quantities vary in a manner 

 analogous to the known process of passing from the general 

 integral to the particular solution of a common differential equa- 

 tion. It is evident that the lines of motion in the tube are 

 either directed to centres, or to focal lines, situated on its axis, 

 the positions of which are given for given points of the tube. 



