Mr EARNSHAW, ON FLUID MOTION. 215 



Now putting ,p for the integral of tlie above differential of 

 x, y, z, t, we have 



pu = d^cp, pv = d,j^, pw = rf,0, + a'p = d,,p : 

 and consequently the equation of continuity becomes 

 dr(l> ± a' {d/<p + d/ip + rf/0) = 0, 

 which wiU furnish two cases, according as a" is positive or negative. 



Secondly. We may limit ourselves to those cases in which the 

 proper motions of the elastic medium are very small: which limitation 

 wUl enable us to neglect d^p, d,p, d^p, inasmuch as the variation of 

 density produced by such small changes of the relative position of 

 the particles, will be too trifling to require attention when multiplied 

 by the small quantities u, v, w ; and consequently the equation of con- 

 tinuity may be written 



dtp , 



— + d,u + d^v + d,w = 0. 



Now as it has been proved* for this case that tuh- + vdy + wdx, is 

 a complete differential ( = d<f> suppose), this will become 



— + dj'cp + df(p + d:-(f) = : 



in which we may write - ^ for ^. as is shewn by all writers 

 on this subject, a' being equal to the fraction 



pressure 

 density ' 



Wherefore by help of this hypothesis, we are able to present the 

 equation of continuity under the following form 



r/,^0 = d' (d;'(p + rf/^ + d;'(p). 

 ' Pontecoulant, Th^or. Anal. Vol. i. p. 1 64. 



