Mr EARNSHAW, ON FLUID MOTION. 221 



uniformly with the velocity a. And perpendiculars let fall upon them 

 from the fixed origin will lie in the seven other parts into which space 

 is divided by the co-ordinate planes, one in each. All the properties 

 which have been shewn to belong to the plane-wave in the last Article, 

 may be shewn to belong also to each of these. 



22. Without further proof it will be sufficiently evident that if 

 we suppose <p to take its most general shape, that is, to be the sum of 

 an infinite number of arbitrary functions, involving all possible values 

 of y; g. It, it would represent an infinite number of independent plane- 

 waves, each transmitted with the same velocity a : and each moving 

 parallel to itself. These waves would be inclined at all possible angles 

 to the axes. This corresponds to the most general form of the integral. 

 Any number of values of ^^ g, h might be omitted in the integral, and 

 then the corresponding plane-waves would be deficient : or, J", g, h 

 might vary according to some continuous law. Two cases therefore 

 ought to be considered, 



1. When f, g, h vary in passing from one function to another in 

 the value of (p independently . 



2. When f, g, h vary continuously. 



The former of these cases has already been shewn to belong to 

 independent plane-waves: but the latter, which is most important, will 

 be considered presently. (Art. 26.) 



23. When the disturbing cause gives rise to a plane-MV&ve, the ex- 

 pression for the state of the medium must be 



<!> = F{f{x - a) +g{y -(i)+h(z-y)- at}. 



Now it has been shewn for this case (Art. 19) that the state of the fluid 

 may be represented by 



<l>=F{fx' +gy' + hz'), 



X, y, z' not involving t, and being measured from an origin situated 

 any where in the plane of origins whose equation is 



fix -a)+g(y-(i) +h{z-y) = at. 

 r F 2 



