462 Mr. POWER, ON A THEOREM IN FLUID MOTION. 



(2) It is also an exact differential when the initial motion is im- 

 pressed by pistons impelled with finite velocities, and acting upon the 

 external boundary of the fluid. 



(3) It differs from an exact differential by quantities of a higher 

 order than the first, when the motions are extremely small quantities 

 of the 1st order. 



In the first case udx + v'dy + w'dz = 0, which may be regarded as a 

 complete differential with respect to x, y, %, of an arbitrary function of t. 

 The general value udx + vdy + wdz is therefore a complete differential. 

 Or, if we please, we may make u = 0, v' = 0, w = in the preceding 

 argument, which does not affect the reasoning, and the conclusion that 

 udx + vdy + wdz is a complete differential will be valid. 



In the second case, if the velocities u, v, w be communicated to 

 any point in the interior by pistons acting impulsively on the surface, 

 it follows from D'Alembert's principle, that the impulses at the surface, 

 in conjunction with — u, — v, - w, &c. in the interior, must be subject 

 to the conditions of equilibrium. 



Consequently if p be the total reactive pressure sustained at any 

 point in the interior of the fluid during the communication of the 

 velocities u, v, w to that point, we have 



-J- = — (udx + vdy + wdz), 

 P 



consequently, udx + vdy + wdz is initially a complete differential, and 



therefore always continues so. 



In this reasoning the instantaneous effects of the accelerating forces 

 X, Y, Z, are omitted as vanishing in comparison with the finite impulses, 

 but they may be supposed to act after the initial motion has been 

 communicated, and udx + vdy + wd% will, by the general theorem, con- 

 tinue to be a complete differential. 



Thirdly, in the case of very small motions, since 



du , dv r dw y 



Tt dx + dt d y + dt d * 



