Mb. POWER, ON A THEOREM IN FLUID MOTION. 463 



differs from — -J- + dV, by quantities which involve the products of 



u, v, w, -T- , &c, it follows that if we limit our approximation to quan- 

 tities of the first order of these small variables, 



du , dv , dw , 

 -j- dx + ~r x dv + -j-dz 

 dt dt * dt 



is a complete differential with respect to x, y, z; and consequently 

 udx + vdy + wdz is a complete differential to the same degree of ap- 

 proximation. 



F ° r if ~dt dx + d~t dy + Hi d% = d 'f( x > 9* *' *)* 

 performing the partial integration with respect to t, we have 

 udx + vdy + wdz = f t d.f(x,y, %, t) 

 = d . f t /(x, y, z, t), 

 since d and f, relate to different variables. 



Trinity Hall, 



May 3, 1842. 



