Conditions of Rectilinear Fluid Motion. 101 
former relating to the arbitrary forces impressed, the latter to 
the arbitrary manner in which the fluid is put in motion, they 
must be separately functions of the single variable of which 
i ~ do 1 (d¢@?? Loa! “ 
the whole quantity A + Fh ab rate dy is a function. 
Consequently no supposition more general than that ¢ is such 
a function can be made. If, however, the motion be steady, 
so that 
= = 0, it is possible 4 and V may be functions of a 
single variable while $ is not so. This can occur when the 
motion is parallel to a given plane in concentric circles about 
a fixed axis, and the velocity is a function of the distance (R) 
from the axis. Let V= ¢(R), and R? = a2? + y?. Then 
uda+vdy= >(R) a) which is not an exact 
differential unless ¢(R) = = Also as the motion is by hy- 
pothesis uniform, the effective accelerative force in passing 
from point to point of a circle of motion vanishes; that is, 
dir , ; : 
a Hence 4 is a function of R, and the equations (9.) 
and (10.) become for this instance identical with equation (7.). 
This is a very particular case of curvilinear motion, and ap- 
pears to be the only exception to the theorem I am arguing for. 
The most important result to which the preceding discus- 
sion leads is, that the arbitrary constant introduced by inte- 
grating equation (7.), cannot be considered a function of the 
time only and not of coordinates, merely because u dx +v dy 
is an exact differential. That it may be so considered, the 
condition also of the independence of the variations d v and dy 
must be satisfied. Hence the general integral of the equation 
Biccatimuitys 52,122 =10,.erltickjequation was formed 
of continuity 75 + Ar ae. which equation was formed by 
merely supposing wd a + vdy to be an exact differential, will 
include cases in which the arbitrary constant above mentioned 
is a function of coordinates and not solely of the time. Let 
us take an instance. ‘The general integral is 
g= F(a + y/—1) +fle—yV—D;, 
whence u =F’ (a9 +yV—1) +f! (@ —yVv — 1) 
v=V —1{P (e+ yV —1) -—f/'\(e@-yV—1)}. 
