210 
will not necessarily be a function of p only, since is not 
necessarily, nor even generally so. Hence the conclusions 
we draw above will not generally hold good, for viscous 
fluids even in the cases where v^o- = 0 or i vV = vQ, where 
P 
Q is a scalar. But if the fluid is incompressible in either of 
these latter cases the theorems hold good. 
Note. — Since writing tlie above 1 have seen a paper by 
M. Bresse, Fonction des vitesses, extension des theoremes 
de Lagrange au cas dflin fluide imparfait (Gomptes Rendus, 
March 8, 1880), in which he seeks to show that if o-= aP 
at any time it will remain so throughout the motion. 
In the investigation, however, the author follows Navier in 
taking — 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 incom- 
pressible, as it would follow that vortex motion could not 
be generated owing to the fluid friction, if for instance it 
starts from rest. 
I should explain the defect in his reasoning thus. Wri- 
ting M for and considering p constant, the equation 
affecting a» is 
+ (Sw V )o- + ^ V = 0 or V V T>t(T + — V = 0, 
P P 
from which M. Bresse concludes that if w is at any time 
zero it remains so. Now this seems to require that for a 
small value of w, should not take a value of an infinitely 
greater order. It has been shown by Maxwell that if w be 
a vector function of any point — represents the differ- 
ence between the value of w at that point and its mean 
value over a small sphere of radius r about the point, and 
therefore I conceive that we can not conclude from the 
above equation that vortex motion may not arise in lines 
or surfaces, but merely that it could not appear in a solid 
foi’in, a form of the existence of which we have no evidence. 
