Conditions of Rectilinear Fluid Motion. 99 
ferential. If (d¢) =udwx+vudy, (da) = (apy) Xdez 
P 
— Ydy, and V? = uw + v’*, the result is, 
do ! 
(dr) + (4.5%) +2 (V9) =O 2. KG) 
which is equivalent to 
drx do = ; = d* =) as gt 
(Getty te Naha gd ayy ee 
Now by reasoning like that applied to the equations of equi- 
librium, it appears from equation (8.) that whenudw + vdy 
is an exact differential, the ratio of dy to d x cannot be of 
arbitrary value unless we have separately, 
dn do AMV 4 
aad dada ‘opde ich . . e e ° (9.) 
dx d? aye 
dy v7 ydt ie 
In this case only the equation (7.) may be integrated from any 
one point of tie fluid to any other, and the arbitrary constant 
to be added is a function of the time only and not of coordi- 
nates. The integrals of equations (9.) and (10.) must evidently 
be identical with that of equation (7.), and therefore with 
each other. To satisfy this condition it is necessary that 
do 
in ai 
is itself a function of x and y; and again, to satisfy this last 
condition, it is sufficient if ¢ be a function of that single va- 
riable, as will appear thus. Let 7, be the variable. Then 
since ¢ = cis the equation of a surface of displacement, it 
follows that r, = c! is also its equation. Hence the surface of 
displacement is that of a cylinder having 7, for radius. Con- 
sequently 7, being always in the direction of the motion, is the 
same as 7 in the equation wd « + vdy = V dr; and because 
outs tO ‘Algito:) 
2 
+ ve should be a function of a single variable, which 
v= he V, as well as 4, is a function of r. Also, as the 
motion is in the direction of the radii, and is the same at all 
points of the cylindrical surface, the effective accelerative force 
in passing from point to point of this surface is nothing. But if 
doc be the increment of a line drawn on the surface, the effect- 
; : di dar , 
ive accelerative force = ra Hence aaa 0, and A is a func- 
H 2 
