Drag of water upon water at Low Velocities. 425 
27 
Also, we have r= T= Jeo 
from which we obtain, after some reductions, 
T= — Pasa (9) 
HRP 
If we introduce into this equation the value of / determined 
by (8), we obtain &, which depends on the torsion only. 
From careful experiments made by means of the apparatus 
described at the beginning of this paper the following value 
has been obtained for the coefficient of “drag” : 
1 
i= 307°057 
From this value of f we can determine the relation between 
the slope of a water-surface and its velocity. We have, for the 
equation of motion of the surface, 
aa 2 ieee 
qe = 9 Sint— fa 3 (10) 
where g denotes the force of gravity, ¢ the slope of the surface, 
and x the distance of any particle from the origin measured in 
the direction of the motion. If v denote the velocity of a 
particle, equation (10) becomes at once 
7 tf =o sini; (11) 
which gives, by integration, 
sin ¢ — fv) = const. (12) 
This indicates that the velocity will increase from zero up to 
the value given 
gsinti—fv=0, (13) 
after which it will remain constant forever. ; 
The final constant velocity given by equation (13) is 
g 80? __ 39.9 9¢ 307°057 sin ¢. (14) 
v7= 
If we express the velocity in feet per second, and call A the 
slope per mile, we find 
v = 1°8726 X Aft. per second; (15) 
which is equivalent to 
v = 30°642 A miles per day. (16) 
Dr. Carpenter has proposed to explain the phenomena of 
ocean circulation by the greater height of the water at the 
equator as compared with that at the poles. 
