228 Mr EARNSHAW, ON FLUID MOTION. 



straight line suddenly expand equally. In this case each surface of equal 

 density is a cylinder with hemispherical ends; the given line being the 

 axis, and the extremities of the line being the centres of the hemi- 

 spheres. 



By resolving each wave-surface into component waves, by drawing 

 tangent planes to the cylinder and hemispheres, and supposing these 

 components transmitted, each parallel to itself with the velocity a, it 

 will appear that at any time each wave-surface is of the form of a cylinder 

 with hemispherical ends. The radius of the cylinder increases with the 

 luiiform velocity a. 



29. It is sufficiently manifest, from what has already been done, 

 that the form of a wave-surface depends only on the form of it at 

 any given moment, or indeed only on its initial form ; its magnitude, 

 but not its nature, depends on the time. This is equivalent to saying, 

 that the nature of the equation of a wave-surface depends only on the 

 form of the given disturbance, while the parameters of that equation 

 depend upon the time. This will sometimes enable us to determine 

 the properties of the wave when it is curved, without employing the 

 general integral 



^ = -LFXfix - a)+g-(!/ - /3) +/«(«- 7) - at, f, g, /,}. 



Having determined the form of the wave-surface upon the prin- 

 ciples of Art. 28, let its equation be ^(a;, y, ss; A, B, C ...) = 0; in 

 which A, B, C ... are parameters depending on the time only ; also 

 let the state of the fluid be expressed by the equation 



^ = y\,{x, y, ss; A, B, C ...) 



Now for any point in a wave-surface = constant ; that is, • any 

 values of x, y, % which make x ~ *'' ^^ make -^ = constant. >J> there- 

 fore can only be made to change in value by writing different values 

 of A, B, C. We may therefore say that \// is a function of A, B, C... 

 only, which are connected by the equation x — ^- ^^ ™^y therefore 

 consider x, y, & as functions of A, B, C ... by virtue of ^^ = 0; and 

 having found the values of rf/</), d,f<p, d/cp in terms of the partial 



