Prof. Challis on the Principles of Hydrodynamics. 445 



with this result, let G(v) = m cos {kv + c) . Then it will be found 

 that 



(j) = 7ncos ■< klv— -p^j +c V. 



Or, putting q fov k j-, and substituting the values of v and /a, 



( / 4e A 



= ??jcos^( ^ — a/ V/ \-\ — 2+^ /• 



By using this first approximate value of ^, and integrating (B) 

 to the second and third approximations, exactly the same expres- 

 sions for ^ and ^i result as those obtained by the former process. 

 Tlius the hypothesis of an axis of rectilinear motion has been 

 shown to be compatible with the hydrodynamical equations, no 

 contradiction having been met with in the foregoing investiga- 

 tion. As this conclusion has been an-ived at by the indications 

 of the analysis prior to the consideration of any arbitrary case of 

 disturbance, it may hence be concluded that the action of the 

 parts of the fluid on each other is such, that there is always a 

 rectilinear axis of motion along which the motion is vibratory, 

 and that all the parts of a vibration are propagated with exactly 

 the same velocity. 



If instead of the function G we had reasoned with the function 

 F, the same results would have been obtained, with the differ- 

 ence only that the propagation of the motion would have been 

 in the opposite direction. Hence as the equation (B) to the first 

 approximation is satisfied by the sum of the values of <^, it fol- 

 lows that when the vibrations are small, two propagations may 

 take place simultaneously along the axis in opposite directions, 



(3.) Hitherto the reasoning has been carried on by means of 

 exact equations, and some circmnstances respecting the motion 

 resulting from the mutual action of the parts of the fluid have 

 been ascertained for velocities and condensations of any magni- 

 tude. The laws of the curvilinear motion which takes place at 

 finite distances from the axis of rectilinear motion, and which, as 

 already stated, does not satisfy the condition of integrability of 

 udx + vdy + wdz, can probably be arrived at only by successive 

 approximations, commencing with terms of the first order with 

 reference to the velocity and condensation. The reasoning in 

 future will be restricted to terms of the first order, so that the 

 equations will be hnear. 



Now it may be proved as follows, that if terms of the first 

 order only be retained, the quantity udx + vdy -\- wdz is integrable 

 for all distances from the axis of rectilinear motion. 



Let the pressure at any point xyz at the time t be «^(1 + a), 

 a- being a small quantity the powers of which above the first are 



