140 MR. GWYTHER ON THE LAGRANGIAN FORM OF 



will not necessarily be a function of d only, since Dtd is 

 not necessarily nor even generally so. Hence the con- 

 clusions we draw above will not generally hold good for 



viscous fluids, even in the cases where V^o'=o or -v^cr = 



V Q; where Q. is a scalar. But if the fluid is incompres- 

 sible, in either of these latter cases the theorems hold good. 



Note. — Since writing the above I have seen a paper by 

 M. Bresse, " Fonction des vitesses, extension des theoremes 

 de Lagrange au cas d'un fluide imparfait '' {' Comptes 

 Rendus/ March 8, 1880), in which he seeks to show that, 

 if cr=VP at any time, it will remain so throughout the 

 motion. In the investigation, however, the author follows 



Navier in taking ^ as an absolute constant of the fluid. 



This, of course, will not lead to correct results for a gas, 

 at any rate ; and I think his result must be wrong for any 

 liquid, even incompressible, as it would follow that vortex 

 motion could not be generated, owing to the fluid-friction, 

 if, for instance, it started from rest. 



I should explain the defect in his reasoning thus : — 

 Writing g for VVo":, and considering d constant, the 

 equation affecting g is 



D^§+(S§V)o- + ^V"?=o, orVvD<o- + ^V'?=o, 



from which M. Bresse concludes that, if g is at any time 

 zero, it remains so. Now this seems to require that, for a 

 small value of g, ^^g should not take a value of an infi- 

 nitely greater order. It has been shown by Maxwell, that 



if § be a vector function of any point, V^gi'epresents 



the difference between the value of g at that point and its 



