452 Prof. Sylvester on the Motion and Rest ofFluids* 



-//>.,....(ff-«.)=o. 



Here it must be remembered that <», <|>, ^|/ are perfectly inde- 

 pendent of each other. Also the values of the three first 

 vi^ritten quantities depend upon the values of X^ Y, Zy at the 

 bounding surface ; the values of the three last-vi^ritten depend 

 upon the general values of Xy Y^ Z^. It is clear therefore 

 that each system of three equations and each member of each 

 system must be separately zero. 

 The three latter equations give 



dy dx 



dYj 



dz 



dy 

 d^ 



- 0. 



(y-) 



07^ 



dx z 



The three former require that for each section of the sur- 

 face parallel to the plane x y 



f^iXidx + Yi'^dy) =0 

 for each section parallel to y z, 



fai{Y,dy + Z,dz) =0 

 for each section parallel to z x^ 



r(p{Z^dz + Xy<?/r) =0. 



and these equations are to hold good whatever i>, </>, &> may 

 be. From the equations (y.) we derive 



> 



(S.) 



X.dx + Y^dy -V Z^dz=d^ 



(5.) 



from equations (8) we obtain 



4> = constant for all points in any section of the bounding 



surface parallel to the plane o^ xy 

 ^ — constant for all points in any section of the bounding 



surface parallel to the pLane of ^^ ^r 

 4» = constant for all points in any section of the bounding 



surface parallel to the plane of z x. 

 * See remark at Introduction. 



