OF THE MOTION OF FLUIDS. 197 



But from (A), 



pdt pdr ' dr dr' dr \r r + l) ' 



And differentiating (B) with respect to space only, 



^1^ = Xdx+ Vdy + Zdx-d.^ - VdV. 

 p at 



If the variation be from one point to another in the direction of the 



motion, dx = dr, dy = - — — dr, dz = dr. Hence, 



r ^ r r 



a\dp ^ X ^-° , Y y~^ + Z ^^^ _-^ d(p d'(f> 

 pdr ' r ' r ' r drat dr ' di^ ' 



Substituting this value of — ^ in the foregoing equation, and then 



equating the two values of ,'] , we shall obtain, 



/ d£\d^_Q^ d^ d^t . ,.d^(l , J_\ 



\ ~ dt^j dr' dr ' drat df "*" dr\r "^ r + l) 



+ ^ (x^^ + ry^ + Z'-^) =0 (C.) 



dr \ r r r I 



This is an equation of general application. If, as a particular case, 

 I, a, /3, 7, each = 0, we shall have the equation I obtained in my 

 former paper (Art. 4.) by assuming ^ to be a function of v^a^ + y^ + s!^ 

 and t in the equation {n) of the Mecanique Analytique (Part II. 

 Sect. XII. Art. 8.) 



It may be proved as in Art. 11, that -^ = /~77 ^*' ^^ ^'^^ incom- 

 pressible fluids, and that the equation applicable to steady motion is, 



a' h. 1. p = fiXdx + Ydy + Zd%) - ^ + fit) . 

 Vol. V. Part II. Cc 



