the Vibrations of an Elastic Fluid. 137 



Since, however, the result was not deduced from the combina- 

 tion of all the three general equations, we are not entitled to 

 conclude that all motion which depends only on the mutual 

 action of the parts of the fluid is rectilinear, but that rectilinear 

 motion depending on such action is a law of the motion of an 

 elastic fluid. 



16. As this law of rectilinear motion is of prime importance, 

 it will be worth while to endeavour to place in as clear a light as 

 possible the reasoning by which it was deduced. In art. 11 the 

 line s is denned to be an orthogonal trajectory between a given sur- 

 face of displacement from which the trajectories originate, and any 

 other surface of displacement ; and it was shown that (ds)=0, if 

 the differential be taken from point to point of the latter surface. 

 Now possibly it may be thought that ds = Oj at a given time, 

 merely because the lines s are by hypothesis perpendicular to 

 that surface, and that the parallelism of the surfaces of displace- 

 ment does not flow from this result. The following simple 

 instance will serve to clear up this point. Let the surfaces of 

 displacement be planes passing through a fixed axis. Then the 

 orthogonal trajectories will be circular arcs; and if be any 

 finite angle made by two of the surfaces, we shall have, at the 

 distance r from the axis, s = r0 } and, at the same time, at the 

 distance r -f dr, s + ds= (?• + dr) 6. Hence ds = 6dr ; and s cannot 

 be constant in passing from one trajectory to another more 

 distant from the axis, so long as the surfaces are inclined to 

 each other. 



1 7. These preliminaries being settled, we have next to inquire 

 what particular rectilinear motion satisfying the condition of the 

 integrability of udx-\-vdy-\-wyz the fluid admits of, indepen- 

 dently of the character of the arbitrary disturbances. For 

 instance, that function would be integrable per se if the motion 

 were in lines perpendicular to a plane. But when we have ob- 

 tained, by strict analytical reasoning, equations derived from the 

 hypothesis of such rectilinear motion, we find that they give 

 contradictory results, and admit of no interpretation relative to 

 the motion of an elastic fluid. I allude to the known equations 



w =■ a Nap. log p =f(z — (a + w) f\ 



from which, as I have shown in vol. xxxii. S.3. of the Philosophical 

 Magazine (p. 496), the inference may be drawn, that points of 

 no velocity and condensation may at the same instant be points 

 of maximum velocity and condensation. This contradiction 

 merely shows that the rectilinear motion which is the subject 

 of the present inquiry (that, namely, which is due to the mutual 

 action of the parts of the fluid) is not of the supposed kind. 

 If no contradiction had been met with, we must have concluded 

 Phil Mag. S. 4. Vol. 24. No. 159. Aug. 1862. L 



