Differential Equations applicable to the Motion of Fluids. 85 



that the partial differential equations of fluid motion (namely, 

 those in which the principal variable is usually designated by 

 <fi) cannot be generally applied, unless the variation of the co- 

 ordinates from one point to another of space at a given time, 

 be restricted to take place in the direction of the motion. The 

 necessity for this limitation, which might reasonably be ob- 

 jected to if the equations were general, arises from the limited 

 nature of those equations. Were we in possession of the most 

 general equations no such limitation would be required. In 

 support of this view 1 proceed to point out the method of find- 

 ing the general partial differential equations of fluid motion. 



Let the pressure [p) and density {p ) of the fluid be related 

 to each other by the equation j) = k p. Let u, v, w be re- 

 spectively the resolved parts of the velocity, in the direction 

 of the axes of coordinates, of a particle whose coordinates 

 at the time t are or, j/, ;:. Putting for shortness sake P for 

 k . Nap. log p, and supposing no extraneous force to act, we 

 have the known equations 



dp d.pu d 

 dt dx dy 



To render these equations applicable to instances of fluid 

 motion, it is requisite to transform them into others contain- 

 ing partial differential coefficients of a single variable. This 

 has been done in the case in which ud x -{-v dy -k-tsodz is an 

 exact differential, with respect to space, of a function of a?,?/, ^r, 

 and ^, by substituting d <|) for this quantity. The views I have 

 detailed in the Number of this Magazine for last June (S. 3, 

 vol. xviii. p. 477), led me to infer that this is only a limited 

 case, and that in every instance of fluid motion essentially 

 different from that of a solid, ud x + v dy + ixs dz may be 

 made integrable by a factor. Independently of those views, 

 it is clear that the reasoning is not conducted in the most 

 general manner by supposing that quantity integrable per se, 

 and it is remarkable that no one hitherto has traced the con- 

 sequences of introducing a factor. 



Let us therefore suppose -^ to be the factor, and let 



-^dx + -^dy+-^dz = d. <p{x,y,z,t). 

 I have already proved (Philosophical Magazine for Sep- 

 tember, p. 230) that ^-dx + ^-dy + ^- dz = '\s the 



