102 Prof. Challis’s further investigation of the 
Suppose F’ (2+y/ —1)= F(a +y V =1),andf'(e@—y 7 1) 
=F (@-yv = 1). Thenu=ma,v=—my, and $* = 0. 
Ly Os a) 
Taking y and 2 coordinates of a line of motion, 
a ‘ 
QE. Shea ae 
Hence x y = ¢, is the equation of the lines of motion, which 
are therefore hyperbolas. The equation giving the pressure 
becomes, when there is no impressed force, 
p=C— me? +7’). 
If now C be a function of the time only, we shall have p = 0 
at all points of the cylindrical surface of which the equation 
is C = m (x? + y?), and beyond this surface the fluid does not 
extend, because the value of p beyond this limit becomes nega- 
tive. The boundary of the fluid is therefore at all times a 
cylindrical surface. But if this be the case at one instant, it 
cannot be so the next, since the velocity is the same at the 
same time at all points of the surface, but differs in direction. 
This contradiction arises from assuming C to be a function of 
the time only in a case of curvilinear motion for which ud « 
+ vdy is an exact differential. 
It is of so much consequence to establish fully the conclu- 
sions I have now arrived at, that I shall be excused for ad- 
verting to the arguments of other writers which appear to 
militate against them. A paper by Mr. Stokes on the steady 
motion of incompressible fluids in the recently published part 
of the Transactions of the Cambridge Philosophical Society 
(vol. vii. part 3. p. 439.), contains a proposition, which, if it 
be correctly proved, is a direct contradiction of the views I 
have here maintained. The instance (in p. 439.) is that of 
the uniform motion of fluid, issuing from an orifice in a vessel 
containing an indefinite expanse of fluid. The equation ap- 
plicable to this case, supposing the motion to be parallel to 
the plane of wx y, is 
P=Q—-—iw+wv)+C, .... (il) 
where dQ=Xdax+Ydy. If the fluid were at rest we 
should have p, = Q + C, and C, would be absolutely con- 
stant. In the case of motion C may be determined by con- 
sidering the pressure and velocity at any point indefinitely 
distant, where p is indefinitely near to p, and u and v are in- 
definitely small. Hence C is indefinitely near to C,. Mr. 
Stokes goes on to argue that C is therefore equal to C, and 
