Mr. Woolhouse on the Deposit of Submarine Cables. 347 
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of cable, we should have that length = ZT. Therefore for the 
accelerative forces on the element ds due to the change of tension, 
we have 
! 
Horizontal force =g s( T cos ») = = (T cos w), 
Vertical force =g Z (¢ T sin ») =i - (T sin @). 
By resolving the resistance and friction, the total impressed 
aecelerative forces are therefore,— 
Horizontal = g' = (T cos w) —bn sin? w + b!(m—n cos @) cos w, 
Vertical =a! = (Tsinw) —g'+dnsinwcosw+b!(m—ncos@)sina. 
Now if we suppose the movement of the cable to be steady, 
so that it shall be uniformly deposited at the same speed at 
which it is delivered, and such that the suspended portion shall 
retain its form, the absolute velocity of a definite point of the 
cable will evidently result from a movement down the curve at 
the velocity of delivery taken in combination with an onward 
movement of translation équal to the ship’s velocity. Thus we 
have 
a a d*z' ___,,@c0s@ _ od cosa 
i? a ne ee aa eco, cae. 
dy _ : d*y dsinw _,dsinw 
Fe ets deme aoe. dita eos dt = ib . 
It may be proper to observe that these conditions of steady 
movement, on which the investigation is made to depend, can 
only be accurate when the cable is paid out at the same speed as 
the ship’s velocity, or when m=n. When this is the case, it is 
evident that, whatever may be the initial disturbances, the move- 
ment will soon cause the cable to assume the permanence of form 
here supposed. The conditions will, however, approximately 
subsist when m differs but slightly from n, as the form of the 
curve will then only be subject to a very gradual change. But 
if m should differ much from n, we ought not to place much 
reliance in the accuracy of the results*. 
dz! dy 
dt?’ dt? 
* The discussion of the problem, taken in all its generality, supposing 
the form as well as the position of the curve to vary, would lead to expres- 
sions too complicated to be of any practical utility. Besides, this is not 
needed, as all irregular movement should be avoided during the operation. 
Equating the values of with the impressed forces, 
