Mk EARNSHAW, on fluid motion. 225 



we shall obtain all possible functions, and amongst others 7 functions 

 similar to those mentioned before (Art. 21). This function, therefore, 

 may be considered as the representative of all, and consequently <p is 

 now properly represented by 



cp = ^F\f{x -a) +g(y-^) +h{z- y) - at, f, ff, /,J. 



This equation is tiie representative of any continuous wave, whatever 

 be its form. The integral of tiie equation of continuity furnishes no 

 other general means of representing a continuous wave. Hence, what- 

 ever be the nature of the original disturbance which gives rise to a 

 continuous wave, the above equation teaches, that the wave may be hy- 

 pothetically resolved into an infinite number of plane-waves moving 

 with the same velocity a. 



Plane-waves are therefore shewti to be the proper components of curvi- 

 linear waves. 



And since the values of/ g, h are the cosines of the inclinations of 

 these component-waves to the co-ordinate axes, if the original disturbance 

 produce a single wave, / g, h will follow some Icm, that law being in 

 fact the condition that the original disturbance may be single. 



The disturbance being thus resolved into component plane-waves, 

 each component is to be supposed transmitted parallel to itself with' 

 the velocity a, and at any time we may compound them into a single 

 wave, by finding the surface to which they are all simultaneously tan<vent 

 planes. " 



27. In confirmation of these views, I shall make a few observations 

 upon general and singular solutions of common differential equations of 

 two variables, the theory of which is weU understood and allowed. 



Suppose, for instance, it were required to find a curve, such that the 

 rectangle of the perpendiculars drawn from two given points upon any 

 tangent shall be constant. This problem produces a differential equation 

 whose general solution is 



y ^-fx± \/nf' + h\ 



