THE EQUATIONS OF FLUID-MOTION. 141 



mean value over a small sphere of radius r about the point ; 

 and therefore I conceive that we cannot 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 form, a form of the existence of which we have no 

 evidence. The true criterion would be found by equating 

 to zero the expressions found above. 



Although I conceive that the theorem is not proven for 

 the case which M. Bresse considers (where //. " may have 

 any value other than zero)/^ I think that i£ jx is sufficiently 

 small a proof may be given. For if /x were zero the pro- 

 position is true, and the terms owing to which it departs 

 from the truth will appear with ft as a factor, and may 

 therefore be omitted from the term (/.y^g ; and under this 

 hypothesis, if j«,v^§ is ever zero it will continue so, and the 

 proposition is completed. 



III. 



Considering how the portion S,V<Pi+SzV<Pi + ^3 VPj of 

 the velocity can be impulsively generated, we see that the 

 initial equation of motion will take the form 



f'T /»T J 



where T is the infinitely small time during which the im- 

 pulsive force rj/ acts, and p is the impulsive fluid-pressure. 

 Now generally there will be no impulsive forces acting 

 bodily on the fluid. But the velocity will be generated by 

 the impulsive pressures only ; and therefore, if a does not 

 satisfy Y^cr = o, it must be on account of one of two 

 reasons : either during the impulse d does not follow the 

 law of dependence on p, which is highly probable ; or j) is 

 discontinuous, so that the form \/p is an improper form. 



