XXI. A general Investigation of the Differential Equations applicable 

 to the Motion of Fluids. By the Rev. James Challis, M.A., 

 Plumian Professor of Astronomy and Experimental Philosophy in 

 the University of Cambridge. 



[Read April 11, 1842.] 



1. Let p be the pressure at any point of a mass of fluid in 

 motion, the co-ordinates of which referred to three rectangular planes 

 are x, y, «, at a time / reckoned from a given epoch; let p be the 

 density at the same point and at the same time, and suppose p and 

 P to be always related to each other by the equation p = of p. Let 

 X, Y, Z, be the forces impressed in the directions of the three rect- 

 angular co-ordinates on the fluid particle which is at the point xy% 

 at the time t, and let u, v, w, be the components of the velocity 

 of the particle in the same directions. Then the two fundamental equa- 

 tions of Hydro-dynamics are, as is well known, 



dp d.pu d.pv d.pw _ 



dt dx dy d% ~ ^ ' 



And we have also, 



dx dy d% 



U= dt> V= di' W =di- 



It will be proper to explain here, that in the above equations, 

 and in the subsequent investigation, the following notation is, adopted 

 for the sake of perspicuity. The differential coefficients of the quan- 

 tities p, p, u, v, w, are partial when they are not in brackets ; when 



