187, 188] HAMILTON'S PRINCIPLE. 501 



*l 

 J 



-dt 9 Sx + Sy + | i, - (Xix 



The integrated terms vanish as usual at the limits for the time. We 

 may now integrate the last three terms with respect to the space 

 variables , obtaining 



= / IP ( x cos ( nx ) + $y cos ( n y) + d cos 



fff &.'*+%+ 



+ 



If we assume that dx, dy, d# vanish for the particles of the 

 fluid at the bounding surface, the surface integral vanishes. We 

 therefore have, collecting the terms according to 8x, dy, dz, 



23) 



By the usual reasoning the coefficients of dx, 8y, dz must vanish, 

 giving us the equations of motion 6). 



188. Equation of Activity. Subtracting from both sides of 

 the first equation 6) the quantity 



9 N du . dv . dw 



we obtain X1 



3 / 2 



" 



^ du , /du dw\ /dv du\ 



25) ^T + ^(^ o- v(o -- TT- 

 ' ^^ ' w dx) \dx dj 



TT- 



dy 



If the applied forces are conservative and derived from a potential F, 

 the right-hand member is the derivative, 



