220 Mr EARNSHAW, ON FLUID MOTION. 



and therefore the state of the fluid particles composing that wave- 

 surface, is represented by 



</. =F{ii, Jig; h). 



Wherefore the state of the particles composing a wave-surface is 

 a function only of its distance from the plane of origins. By 

 varying jo we pass from one wave-surface to another; and a correspond- 

 ing change takes place in the value of F{p, f, g, h) or <p ; the degree 

 or law of this variation in (p entirely depends upon the form of the 

 function F, which is arbitrary. If now we enquire what is meant by 

 the fact of (p being denoted by an arbitrary function, it is clear we 

 learn from that arbitrariness, that wave-surfaces may follow each other 

 according to any arbitrary law of sequence ; if that law be continuous 

 (that is, if F be a continuous function) it will furnish us with an 

 infinite number of wave-surfaces following close upon each other and 

 composing a simple ivuve, due to an original single disturbance of a 

 continuous nature. If 7^ be a discontinuous function, the law of se- 

 quence of the wave-surfaces will be discontinuous also, and therefore 

 in this case we shall have a wave composed of the broken parts of 

 several waves, joined together, or separated by finite intervals, as the 

 case may be. All this will be very evident, by considering p as the 

 abscissa, and (p, which represents the state of the medium, as the or- 

 dinate of a curve whose equation is 



cp = Fip, /, g, h). 



Upon the whole, then, equation (1) represents a plane-wave, (made 

 up of plane wave-surfaces), which is uniformly transmitted in a di- 

 rection from the origin with the velocity a, unchanged in all respects, 

 through the medium; in such a manner that it continues parallel to 

 itself during the transmission, and the perpendicular upon it from the 

 fixed origin falls within that part of space where x, y, s are all positive. 



21. By similar investigations which it is not necessary to enter upon, 

 it may be shewn that, the seven other functions involved in the expres- 

 sion for (p, denote respectively seven plane waves, each transmitted 



