294 CELESTIAL MECHANICS. I. vii. 33. 



d> ^ d'a^ d'a„ d'«- d'u ^ d'«„ 



-H^S. (314, and xg (^)^=„S.£|+ u^ g-fuSz 



-r — ; and the variations of the other parts of the 



expressions being transformed in a similar manner, the 

 sum will obviously be equal to the corresponding terms 



of (H) ]. The fluent of this equation is F-/-^ = ^ 



+ i { i^^y-^ilff-^i^y}' Itwouldbenecessary 

 to add an independent constant quantity, expressed in 

 terms of t, to this fluent, but this quantity may be sup- 

 posed to be included in ^. The velocity of the particles, 

 in the directions of the coordinates, is obtained from the 



d> dV , d> 

 quantity <p; smcewzi-- » vzh—L^ and w=-r-^. 

 ax ay az 



The equation (K), expressing the continuity of the fluid, 



or0=-^+---^+--7i-^ + —^, becomes Ozz-^+'-^ . -^ 

 at ax 01/ dz at ax ax 



. dV d>^d'p d> ^ /dd> , dd> , dd>\ ^, 



-f-i.-rH--i._-l+ o (__r+.^+.^); consequently, 

 dydy dz dz ^ \ dx^ d/ dz^ / ^ ^' 



with regard to homogeneous fluids, since df =0, we have 



dd> dd> dd> r _ d ^ d't; d'l^ -i 



""d^ "d/ dz2 L"~dx dy dz J' 



369. Theorem. If the quantity w j? + 

 t;g7/ + zi^^z is an exact variation of a^^y^ and ;2^, at 

 any one instant, it will always remain so. 



If, for example, this variation be at any one instant 

 equal to ^(p: it will be at the next instant equal to d^+Ai 



( zif 3x + -7^ 3VH — T— ^z) which will still be an exact 

 \dt dt at f 



