214 Mb EARNSHAW, ON FLUID MOTION. 



I. Motion cannot be represented by one of the functions alone, 

 which enter into the value of (p. 



IT. Under certain conditions a disturbance may be transmitted 

 through a medium unchanged in form and intensity, and the velocity 

 and direction of transmission seems to be arbitrary : that is, to depend 

 on the manner in which the disturbance is excited. 



III. In other cases the velocity and direction of the transmission 

 may be arbitrary as before ; but the form and intensity of the dis- 

 turbance will undergo continual change with the time. 



IV. The proper motion of every particle is in the direction of a 

 normal to the wave-surface in which it is situated. 



ON THE MOTION OF ELASTIC FLUIDS. 



14. The general equation of continuity for this case is 



d,{pu) + dyipv) + d^ipw) + di{p) = 0. 



But as it is impossible to enter upon the discussion of this equation 

 in the general state in which it now stands, we shall, as in Art. 6, 

 be under the necessity of introducing some hypothesis. 



First. We may suppose the expression 



pudx -j- pvdy + ptvdx ± a^pdt, 



a complete differential of x, y, x, t: in which case 



p(udx + vdy + wdx), 



will be a complete differential of x, y, %: and therefore 



udx + vdy + tvdz 



will be integrable by a multiplier : wherefore the properties of 

 wave-surfaces proved in the first five Articles will be applicable to 

 this case. 



