Conditions of Rectilinear Fluid Motion. 103 
absolutely constant; and differentiating on this principle 
equation (11.) with respect to x only, he obtains 
dp du dv 
epee Bey SG a Py Ste ne ON is oct (12.) 
ay & Diy du du : 
Also Fie ae taamiai, wale dy’ by the usual equation. 
Hence by subtraction, v (<2 — £2) =O) 
AY i Ae 
To this reasoning it may ve objected, that when C is assumed 
to be absolutely the same as C, there is no longer any motion ; 
the fluid is at rest, and the last equation is satisfied, not be- 
du_dv 
sine Yi ae 
of motion differs from a state of rest, C differs from C, by an 
infinitesimal quantity of some order; and as this quantity, 
however small, is a function of the coordinates of a line of 
motion, it is not allowable to differentiate equation (11.) with 
respect to # only. The equation (12.), being obtained on a 
false principle, is not even approximately true. An example 
will illustrate this. Let the fluid descend by the force of gra- 
vity (gy), between two planes perpendicular to the plane of wy, 
and inclined to each other at a given angle, and let the motion 
be parallel to the plane of ay. Suppose the lower surface of 
the fluid to be always bounded by a horizontal plane to which 
an arbitrary motion is given, and the upper free surface to be 
kept horizontal by the force of gravity. An instance of fluid 
motion similar to this I have considered in the Philosophical 
Magazine for January 1831, and in the Cambridge Philoso- 
phical Transactions (vol. v. part 2. p. 186.). By reasoning 
as in that instance, I find that the velocity is in straight lines 
directed to the intersection of the two planes, and in a given 
line varies inversely as the distance from the intersection. 
Also that the vertical velocity is the same at all points of the 
same horizontal plane. Hence, taking x and y vertical and 
horizontal distances from the intersection of the planes, if U 
be the vertical velocity, supposed uniform, at the vertical 
Uh Uhy 
Ui ‘ho how 
x 
but because wand v each = 0. Since a state 
distance i, we shall have vu = » and 
U? h2 2 . 
p=C-gx->7(1+4), out oe) (emia (i3,] 
the integration being performed along a line of motion. If 
