Mr EARNSHAW, ON FLUID MOTION. 219 



When referred to this new origin, the state of the fluid is repre- 

 sented by the equation 



<p = Fijx +gy' + hz', f, g, h) (3). 



Now the co-ordinates of the moveable origin are X, Y, Z, amongst 

 which quantities there is no relation but that which is expressed by 

 equation (2); wherefore, though the origin must be some where in the 

 plane whose equation is (2), its position in that plane is perfectly 

 indeterminate. And what point soever in this plane we take for 

 origin the state of the fluid in reference to that point is expressed 

 by (3), an equation which does not involve /. Hence the state of the 

 fluid IS invariable with respect to the plane (2) ; i. e. to whatever 

 point m this plane we transpose the origin, the same values of x y ^' 

 will give the same value of ^, or denote a point in the same ^'^.e- 

 surface. Hence the wave-surfaces denoted by (1) are always parallel 

 to tlie plane (2), and preserve an invariable distance from it Conse 

 quently the wave denoted by (1) is a plane wave travelling parallel 

 to and at the same rate as, the plane (2), which may be called the 

 p/ane of origins. 



Now because f^ + g'--, /f = i, ^ ^, u denote the cosines of the 

 angles of inclination of a line to the axes of co-ordinates; the line 

 Itself IS, as IS well known from the principles of Analytical Geometry 

 perpendicular to the plane (2), and if drawn from the ori^^in of the 

 co-ordinates X, Y, Z, that is, from the original origin fixed in space 

 Its length IS equal to at. AVherefore the plane of origins travels in 

 such a manner that at is the length of the perpendicular upon it 

 from a fixed point: and consequently it moves with the uniform 

 velocity a. As the wave-surfaces are always paraUel to it, and preserve 

 their distances from it unchanged, they are transmitted through the 

 medium with the same uniform velocity a. 



20. If the distance of a wave-surface from tiie plane of origins 

 be denoted by p, *' 



P =fx' +gy' + hx. 

 Vol. VI. Paut II. Ff 



