222 Mk EARNSHAW, ON FLUID MOTION. 



Now this equation denotes a plane of infinite extent, and as the wave- 

 surface may be referred to any point in it, without altering the equation 



(p = F {fx ^ gy' + M), 



which expresses the state of the fluid, the same values of x, y, x, 

 which, measured from a given origin, give the value of at a dis- 

 turbed point, will give, when measured from every point in the plane 

 of origins, a series of other points in the fluid, at which the value of 

 <p is the same as before; therefore the plane of origins and the wave 

 would seem to be of equal extent. What are we then to infer from 

 this circumstance? The plane of origins is manifestly infinite, but 

 the wave cannot stretch out beyond the limits of the medium. As far, 

 however, as the fluid extends, so far we can prove the wave-surface to 

 extend ; for nothing prevents the equation 



^ = i^(/r' + gy' + M) 



from being true for an infinite plane, but the fact of there being, beyond 

 certain limits, no fluid medium: so far, therefore, as there is fluid, so 

 far the wave-surface extends. Hence, 



A plane-wave cannot he transmitted through any fluid unless it extend 

 completely across the medium, from houndury to boundary. 



Hence, if the medium be divided into two parts by a fixed screen, 

 in which is a finite aperture, the fragment of the wave-surface (supposed 

 parallel to screen) which passes through the opening cannot continue 

 plane ; but it must somehow or other abut upon the back of the screen. 

 What is the precise form which it takes is very difficult to determine. 

 The problem, however, does not seem to be absolutely impossible ; and - 

 appears to depend entirely upon the discovery of some means of- intro- 

 ducing the condition of a limit to the medium into the mathematical 

 expression for <p. 



24. I think it is sufficiently clear that the expression for a plane-wave 

 necessarily supposes the extent of the medium to be infinite. When, 

 therefore, the fluid is of finite extent, we must suppose it infinite. 



