206 Mr EARNSHAW, ON FLUID MOTION. 



differential, it will be so as long as the motion lasts; this is the 

 mathematical expression of the following physical fact; — 



If, at any one instant, the motion of the fluid be ifi wave-surfaces, 

 each surface will travel unbroken through the medium independently of all 

 the rest ; that is, as if the others did tiot exist. 



Or, in other words, if the motion at any one instant be in wave-lines 

 (Art. 2), then the motion can never resolve itself into wave-surfaces; 

 and, conversely, if the motion at any one instant be in wave-surfaces, 

 it can never break up into wave-lines. 



5. If it happen that a particle be situated in two or more wave- 

 surfaces at once, either the particle must be at rest, or the surfaces 

 must have a contact at that point ; for, if in motion, its direction 

 must be perpendicular to all the wave-surfaces. 



However complicated the motion of the fluid may be, it will 

 always take place either in wave-lines or wave-surfaces. For the former 

 will be the case when udx + vdy + wd% is not integrable per se or by 

 a multiplier, and the latter when this expression is integrable. 



Some of these remarks are illustrated in the following example. 



Ex. Suppose the motion of the fluid to be such that 

 udx + vdy + wdz = w'^ {ydx — xdy). 



In this case the differential equation of the wave surfaces is 



ydx - xdy = 0; 

 and therefore, y =f(t) . x 

 is the general equation of a wave-surface in such a motion of the 

 fluid. 



Hence, all the wave-surfaces are planes passing through the axis 

 of z, and the motion of the particles, being at right angles to them, 

 will be in circular arcs parallel to the plane of xy. 



All the particles in the axis of » will be at rest, for there the 

 wave-surfaces intersect each other. 



