XXIV. On the Truth of the Hydrodynamical Theorem, that if udx 

 +vdy+wdz be a Complete Differential with respect to 

 x, y, z, at any one instant, it is always so. By the Rev. 

 J. Power, M.A., Fellow and Tutor of Trinity Hall. 



[Read May 9, 1842.] 



This Theorem was first announced by La Grange, who has given 

 a demonstration of it in the Mecanique Analytique, Tom. n. p. 307. 

 The late celebrated mathematician, Baron Poisson, has, however, in 

 the last edition of his Mechanics, expressed great doubts of its gene- 

 rality, and has even mentioned that examples have occurred to him 

 in which it is in fault. Those examples, however, he has not given, 

 which is much to be regretted, as the theorem is one of the greatest 

 importance in the theory of fluid motion, and if not generally true, it 

 was highly desirable for the prevention of error, that its want of gene- 

 rality should be placed beyond all doubt, which a single legitimate 

 exception would have been sufficient to effect. 



The demonstration of La Grange supposes that the general values 

 u, v, w, the component velocities of any given particle of fluid at the 

 end of the time t, are developable as follows : 



u = u' + u"t + u'"t" + &c. 



v = v ' + v "t + v'"t* + &c. 

 w = w' + w"t + w'"f + &c. 



and Poisson objects that the demonstration fails when u, v, w are not 

 developable in series of the above form, as may occasionally happen. 

 The objection is a fair and reasonable one; and it is my object in 



3F2 



