Prof. Sylvester on the Motion and Rest of Fluids. 451 

 Finally, Gauss's principle teaches us that 



y/y^^.c/y.^^. j^X,AX, + Y,AY, + Z;AZ^J> = 0...(/3.) 



Now 



^(X+X,) ^ ^(Y + Y,) _^ d{Z + Z) 

 dx dy dz 



\dx) "^ \dy) "^ \dz) 



, 2 f^ dw disD du du drT^ 

 ' \dz ' dy d X ' dz dy ' dzj 

 as appears from the ="13 (i) (2) (3) (4), and hence 



f^AX, dAY. , dAZ. 



■ -j — ' + ' = 0. 



dx dy dz 



The complete solution of which, free from the sign of inte- 

 gration, is A Xy AY; A Z^, being subject to no other restric- 

 tions than such as are imposed by this equation 



' d y d z 



, -XT d(o d'\> 

 A Y,:= - 



dz dx 



A Z = — — — 



^'~ dx dy 



ft), (p, vl/ being any three independent functions of Xf y, z. 

 On substituting these values in =» (|3) we obtain 



This may be put under the form 



+fd^ffdy d... {^(„ Y,)- A(„Z,)} 



2G2 



