The true criterion would be found by equating to zero the 
expressions found above (10). 
Although I conceive that the theorem is not proven for 
the case which M. Bresse considers where fx “may have 
any value other than zero/’ I think that if jx is sufficiently 
small a proof may be given. For if fx were zero the pro- 
position is true, and the terms owing to which it departs 
from the truth will appear with ^ as a factor, and may 
therefore be omitted from the term and under this 
hypothesis if is ever zero it will continue so, and 
the proposition is completed. 
IIT. 
Considering how the portion Si v 0i + S2 v 02 + 2g v 0s of the 
velocity can be impulsively generated, we see that the initial 
equation of motion will take the form 
\hdt + / ~pdt. 
) ^ 0 p 
where T is the infinitely small time during which the im- 
pulsive force -ip 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 does not satisfy 
Vvo- = 0, it must be on account of one of two reasons. 
Either during the impulse p does not follow the law of 
dependence on p, which is highly probable; or p is dis- 
continuous, so that the form vp is an improper form. 
IV. 
The velocity can also be written in a form of simple appear- 
ance. Thus — 
01 ^2 . 03 
(7- jja 
since D<0i = 0i - (S(tV0i) = 0. 
Cl ‘3 
Whence D,o- = - D,0i.g - D^02,^ - 
P - d ' d ' d 
