346 Mr. Woolhouse on the Deposit of Submarine Cables. 
further than is contained in these papers ; and having, moreover; 
considered the subject in its more practical bearings, it is pre- 
sumed that the contribution I am now about to make may be 
considered of sufficient value for publication. 
To avoid ¢onfusion we shall, as far as may be Peereie 
retain the notation and general arrangement of the Astronomer 
Royal, and for present convenience we shall here briefly state 
the principal symbols employed, viz.— 
n the ship’s velocity. 
m the velocity of delivery of the cable. 
zx the horizontal ordinate of a point in the cable curve, mea- 
sured, from the point where the curve touches the ground, 
in the direction of the ship’s motion. 
z' the same; measured from a fixed origin. 
y the vertical ordinate of the same point, measured upwards 
from the bottom. 
s the corresponding length of the curve. 
# the inclination of the curve with a horizontal lite at the 
same point. 
p_ the radius of curvature. bree 
T the tension, as measured by the length T of cable weighed 
in water. 
g (=82°19 feet) the accelerative force of gravity in one 
second. ; 
g' the same when diminished in the proportion of the cable’s 
real weight to its apparent weight in water. 
2 
a= a twice the height due to the velocity m with diminished 
gravity. 
I. We have first to discuss the problem on the hypothesis 
that the resistance and friction encountered in passing through 
the water shall each vary simply as the velocity. 
The bottom of the sea is also supposed to be level, and the 
cable perfectly flexible. 
Assume b the coefficient of lateral resistance, 
b! that of longitudinal friction, 
g diminished gravity 
Then, with respect to an element 6s of the cable, we shall lave 
Normal velocity . =nsin@ 
; ; ; downwards ; 
Tangential velocity =m—ncos@ 
Normal resistance =Jdnsin@ a warld 
Tangential friction =d' (m—ncos@) P j 
Also if the tension were measured by the real weight of a length 
