152 PROFESSOR O. REYNOLDS ON INCOMPRESSIBLE VISCOUS 
This equation does not admit of solution from a state of rest ;* but assuming a 
condition of steady motion such that du/dt is everywhere zero, and dp/d« constant, 
the solution of 
CERO op ON a IUD Eo | 
p 2) i GBF o 
if : 
u = du/dz =0 wheny=+b,, } (34). 
is | 
J 
1 dp y — by 
U 
p da 2; 
This is a possible condition of steady motion in which the periods of wu according to 
Art. 8 are infinite; so that the equations for mean-motion as affected by heat- 
motion, by Art. 8, are exact, whatever may be the values of 
ut, Do, p, , and dp/dza. 
The last of equations (34) is thus seen to be a singular solution of the equations (15) 
for steady mean-flow, or steady mean-mean-motion, when w’, v, w’, p’, &c., have 
severally the values zero, and so the equations (16) of relative-mean-motion are 
identically satisfied. : 
In order to distinguish the singular values of u, I put 
—b 
whence ee aijoMece (BS). 
dp ByT : b?—y? | 
—=———U,, U = —U, 2 = 
dx be Un, 2 Un Eee) 
b 
=U, | TRO —= PA Us ] 

According to the equations such a singular solution is always possible where the 
conditions can be realized, but the manner in which this solution of the equation (1) 
of mean-motion is obtained affords no indication as to whether or not it is the only 
solution—as to whether or not the conditions can be realised. This can only be 
ascertained either by comparing the results as given by such solutions with the results 
obtained by experiment, or by observing the manner of motion of the fluid, as in my 
experiments with colour bands. 
* Ina paper on the ‘Equations of Motion and the Boundary Conditions of Viscous Fluid,” read 
before Section A at the meeting of the B.A., 1883, I pointed out the significance of this disability to be 
integrated, as indicating the necessity of the retention of terms of higher orders to complete the 
equations, and advanced certain confirmatory evidence as deduced from the theory of gases. The paper 
was not published, as I hoped to be able to obtain evidence of a more definite character, such as that 
which is now adduced in Articles 7 and 8 of this paper, which shows that the equations are incomplete, 
except for steady motion, and that to render them integrable from rest the terms of higher orders must 
be retained, and thus confirms the argument I advanced, and completely explains the anomaly. 
