210 Mr EARNSHAW, ON FLUID MOTION. 



II. Sometimes the disturbance may be such as to render it possible 

 to introduce t entirely into the parts f(,x - a) ±g{y - /3), so that the 

 integral may be written 



ct> = F {/{x-a- T) + g{y-(i-r),f,g} 



+ F, {/{x-a- T) - g{y-(i-T),f, g}. 

 T and t being functions of t. 



Let us in this case refer the motion of the fluid to the moveable 

 origin, in the plane of xy, whose co-ordinates are T -V a, t + /3; which 

 will be done by writing x + T+a, y' + t +(i for x and y; then the state 

 of the fluid is expressed by 



<p = F ifx' + gy,f, g) + F, ifx - gy',f, g) ; 



an equation which does not involve t; the state of the medium is there- 

 fore perfectly invariable with respect to the moveable origin. The original 

 disturbance, then, of what kind soever it may be, is transmitted 

 through the medium unaltered in all respects, with a velocity equal 

 to that of the origin of co-ordinates, that is, of the point T + a, 

 T-I-/3; hence the velocity of transmission, in the direction of x, =d,T, 

 and in the direction of y, = d,T. 



III. It may happen that it will be impossible to introduce t 

 entirely within the parts f{x - a) ± g{y - fi); let it however be done 

 as nearly as possible, so that the equation may be written 



^ = F {/{x-a-T)+g{y-[i-^),f, g, f] 

 + F,{f{x-a-T)-g{y-(i-T), f, g, t\. 



After transposing the origin as before, this becomes 



^ = F {/x'+gy, f, g, t} + F,{fx'-gy', f, g, t]. 



Whence it appears that the forms of the equations of the wave- 

 surfaces will remain unchangeable; but t entering into the parameters, 

 shews that the magnitudes of the wave-surfaces will change with the 

 time. In this case therefore the wave-surfaces will be transmitted through 



