Mr. POWER, ON A THEOREM IN FLUID MOTION. 461 



and consequently 



u'dx + v'"dij + w'"dz 



is an exact differential, and the demonstration may be carried on as 

 far as we please. 



Hence it also follows that 



(u'dx + v'dy + w'dz) + (u'dx + v" dy + w" dz)t + &c. ; 

 that is, (udx + vdy + wdz) is an exact differential. 



If then, udx + vdy + wdz be an exact differential when t = 0, it is 

 so when t is very small ; and since the origin of t is arbitrary, by a 

 repetition of the same reasoning we conclude that it is an exact differential 

 at the end of a second very small interval, and so on ad infinitum. Hence 

 we conclude that it is so for any finite value of t. 



Moreover, since the origin of t is arbitrary, and t may be taken 

 either positively or negatively in the preceding reasoning, it follows 

 that if at any one instant udx + vdy + wdt is an exact differential, it 

 is so at all other instants, past and future. 



Hence also it follows that if udx + vdy + wdz is not an exact dif- 

 ferential at any one instant, it never will be so during the whole 

 motion, for if it were so at any other instant, it would likewise be 

 so at the former instant. 



Having thus, as I conceive, supplied the deficiency of La Grange's 

 demonstration ; much as I respect the authority of Poisson, I may be 

 allowed to venture an opinion, that he may have formed a hasty judge- 

 ment on the cases before him, which he has not thought proper to 

 detail, and which seemed to militate against the generality of this 

 theorem. 



I shall conclude with a few remarks in confirmation of the three 

 following consequences of the theorem. 



(1) The expression udx + vdy -f wdz is in all cases an exact dif- 

 ferential when the motion commences from rest. 

 Vol. VII. Pabt III 3G 



