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[ 123 ] 



XX. 0?i the Integration of the General Equations of the Mo- 

 tion of Incompressible Fluids. By J. Challis, Fellow of 

 Trin. Coll. Cambridge, and of the Cam. Phil. Soc. * 



HE theoretical investigation of the laws of motion of in- 

 compressible fluids, conducted in the most general man- 

 ner possible, leads to the equations, 



^^d_^ ^ = lf ^=£f. (3) 



dx dy^ dz 



{Poisson, Traite de Mecawiqne, torn. ii. p. 486.) 



q is the density of the fluid, p the pressure at any point, the 

 co-ordinates of which are x, y, z ; u, v, w, are the velocities in 

 the dii'ections of j:, j/, z, respectively; <:? V = X^Z^r + Y(Zj/ + 

 Tjdz, X,Y, Z, being the accelerative forces impressed at the 

 point ; and f is a function of x, y, z, and t, such that 

 {d(f) = udx + vdy + iiodz. 



Consequently the above equations apply only to cases in 

 which udx + vdy + wdz is a complete differential of x, y, 

 and z. 



Before any use can be made of equations (1) and (3), the 

 function (p must be obtained from (2). This has been effected 

 approximately by the method of series, and the equations 

 have been made available in a few particular instances. It is 

 however certain that an exact integi'al of (2) may be found by 

 putting o:^ + j/^ + s'^ = r ' ; and as every integral must have 

 a meaning, I propose to consider what is the meaning of the 

 integral thus obtained. 



As r^ = x^ +/ + z\'^ ^ ^_i/il = ^f .£. 



"' ' dx dr dx dr r ' 



d'(p d^(p i» , d(p 



I Icncc 



IJut 

 Therefore 



• Communicated by the Author. 



R 2 Integrating, 



