508 Proceedings of the Eoyal Society 
ties, — one equal and parallel to that of the ship forwards, and the 
other obliquely downwards along the line of the cable, equal to 
that of the paying out, obliquely downwards along the line of the 
cable (since if the cable were not paid out, but simpler dragged, 
while by any means kept in a straight line at any constant incli- 
nation, its motion would be simply that of the ship). Hence, if v 
be the ship’s velocity, and u the velocity at which the cable is paid 
out from the ship, we have 
^ = vsin^*, = w — ■^;cos^ .... (2.) 
Now, as probably an approximate, and therefore practically use- 
ful, hypothesis, we may suppose each component of fluid friction to 
depend solely on the corresponding component of the fluid velocity, 
and to be proportional to its square. Thus we may take 
P = w^, Q=w| (3.) 
where p and denote the velocities, transverse and longitudinal, 
which would give frictions amounting to the weight of the cable ; 
or, as we may call them, the transverse and longitudinal settling 
velocities. We may use these equations merely as introducing a 
convenient piece of notation for the components of fluid friction, 
without assuming any hypothesis, if we regard p and (J as each 
some unknown function of p and q. It is probable that p depends 
to some degree on q, although chiefly on p ; and vice versa^ to 
some degree on p, hut chiefly on q. It is almost certain, however, 
from experiments such as those described in “Beaufoy’s Nautical 
Experiments,” that J) and ^ are each very nearly constant for all 
practical velocities. 
Eliminating p and q between (1), (2), and (3), we have 
W cos 
. /v sin t'Y 
9 ), 
which gives 
\ /> 
^ _ V sin i 
Vcos^ 
( 4 .) 
and 
(WD-T)sin 2 =r 
/’u'—v cos 'i \2 
wd( ^ ) . . . . 
(5.) 
which gives 
q = - -y cos i) ^ 
I WD 
V(WD-T)sint • • • • 
(6.) 
