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



NOTE. 



La Grange, having shown that in very small motions — dx -\ dy -\ dz 



dt dt dt 



is approximately a complete differential, concludes briefly as follows : " et l'on voit 



d> u , dv , dw , .. . , , , , 



que — - dw + — dy + — dz devant etre integrable relativement a x, y, x, la quan- 



tite udx + vdy + wdx devra Tgtre aussi." This remark, which the author leaves 

 sufficiently obscure, in the conclusion of my paper I endeavoured to put under a 

 clearer point of view. 



But since the Paper was printed, I am indebted to my friend Mr Stokes, of 

 Pembroke College, for a remark which convinces me that the conclusion is invalid. 



In fact, I had not thought it necessary to exhibit the arbitrary function of 

 x, y, x which ought to be added after the partial integration with respect to t, 

 conceiving it to be implied under the sign f t , but it it clear that as regards this 

 argument, it ought to be exhibited. Thus the partial integral of 



du dv , dw 1 ■ 



~dt d~t y ' + ~dT '^"' y ' *' ^' 



gives udw + vdy + wdx = f t df(x, y, x, t) + u dx + v dy + iv Q dx, 

 « » «o> 2»o being functions of w, y, % without t. 



And the second side of this equation being put under the form 

 dft (*i y, *> t) + u dx + v dy + w dz, 



it does not follow that udx + vdy + wdx is a complete differential unless u dx + 

 v dy + w dz be so. But, in general, there does not appear to be sufficient ground 

 for supposing this to be the case. 



Or we may reason thus, assuming the series for u, v, w, we have 



udx + vdy + wdx = u'dx + vdy + w'dx + t K . (u"dx + v" dy + w"dx) + &c. 



whence — du + — dy + -— ■ dx — X . t x ~' 1 . (u'dx + v"dy + w"dx) + &c. ; 

 dt dt dt 



and since the left-hand side is a complete differential in very small motions, the 

 right-hand side is so, and consequently u" dx + v" dy + w" dx, &c. are so: but we 

 cannot conclude from thence that tidx+vdy + wdz is so, unless u'dx + v' dy + w'dx 

 (i.e., the initial value of udx + vdy + wdx) be a complete differential. 



Hence, except for the above assertion of La Grange, I see no reason to draw 

 any further conclusion for small motions, than has already been drawn for finite 

 motions, namely, that if, for any one value of t, udx + vdy + wdx be a complete 

 differential, it is always so. 



J. POWER. 

 Trinity Hall, Nov. 9, 1842. 



Erratum. Page 458, line 19 correct thus, 'containing t'. 



