of Deposit of a Submarine Cable. S 



with no limitation as to the physical laws. Accordingly the 

 investigation is then undertaken on the suppositions, that the 

 resistance varies as the square of the velocity, that the co- 

 efficient for longitudinal motion' may be different from that for 

 lateral motion, and that the speed of delivery of the cable may 

 be different from the ship's speed. Commencing with the equa- 

 tions in their most general form, it is shown that the straight 

 line is one of the forms of solution ; and the tension of the 

 cable is investigated on different suppositions, as to the propor- 

 tion' of the coefficients, and as to the rate of the ship and that 

 of cable- delivery. I trust that this investigation will be found 

 to possess real practical value. This is the thi7'd part of the 

 paper. 



1 shall now proceed with the investigations. 



1. It is assumed that the bottom of the sea is level, that the 

 ship moves with uniform velocity and delivers out the cable 

 with uniform velocity, and that the form of the curve taken by 

 the cable from the point of leaving the ship to the point of 

 touching the ground is always the same. The movement of 

 any individual point of the cable may therefore be conceived as 

 compounded of two motions, namely, a sliding down of the 

 point along the curve, while that curve travels horizontally after 

 the ship. 



2. The following is the notation which will be employed : — 

 n the ship's velocity, 



in the velocity of delivery of the cable (when it differs from 

 the ship's velocity). 



X the horizontal ordinate of any point of the cable; measured 

 from the place where the cable touches the ground, in the direc- 

 tion of the ship's motion. 



x' the horizontal ordinate of the same point measured from a 

 fixed origin, x' therefore = a; + ?i< + K. 



y the vertical ordinate of the same point, measured upwards 

 from the bottom. 



s the length of the curve from the place where the cable 

 touches the bottom to the same point. 



s' the length of the cable from a fixed origin to the same 

 point. 6-' therefore =s + n/ + L (if the cable be delivered with 

 the same speed with which the ship goes), or = s + 7nt + L (if 

 delivered with a different speed). 



T the tension of the cable at the same point, as measured by 

 the length T of the cable weighed in the water. 



ff the acceleration produced by gravity in one second. 

 a' the acceleration ]n'oduced by gravity when diminished in 

 B2 



