362 Mr. Woolhouse on the Deposit of Submarine Cables. 
Hence as a is a very small quantity and the tension cannot in 
practice exceed the tensile strength of the cable, we conclude 
that, if the depth of the sea be at all considerable, the inclination 
(w); at the ship, must necessarily be very near to the limiting 
angle, especially if that angle be small. 
If we consider the nature of the formula (11), it is evident 
that the quantity under the fractional exponent Trak must 
necessarily be algebraically positive. Therefore cos #—a and 
cos w,—a must have like signs; that is, in any possible form of 
the cable, the inclination » must either be always less or always 
greater than the limiting angle X. Let us briefly examine these 
two cases separately. 
1. When @ is always less than the limiting angle, the values 
of cos #—a will be positive; and therefore the values of T—a 
will be always positive. Also, since the values of cos o —e? sin* 
are positive, it follows from (8) that the radius of curvature is 
always positive. Consequently the curve is everywhere concave 
upwards and convex downwards. 
The equation (12) indicates that the radius of curvature in- 
creases rapidly when approaches to the yalue of A, and that 
the upper portion of the cable rapidly approximates in form to 
that of a straight lime. Also the equation (14) shows that for 
a given value of e, or at a given speed of the vessel, the contour 
of the curve will be given in species, the scale of measurement, or 
unit, being the length of the radius of curvature at the lowest 
point ; and this again will depend upon the amount of “stray 
length ” of cable payed out as compared with the depth of the 
sea. 
Now d(s—2x) ds—dz _1—cos 
iy’ i. | Se 
= 1p, 
= tanzo ; 
.. stray length s—x= {dy tan fa. 00 tenn eS) 
As @ is always less than 2, the superior limit of this integral 
is obviously ytan}2, orc;y. For this extreme limit we must 
have m=; and if this equality exist at any point where the 
tension is finite, it must, by (i1), subsist at all other points; 
and the curve will therefore merge into a straight line. And by 
(14) we have then py=0, and by (3) Ty>=a, which is the least 
possible value of the tension at the lowest point. But as the 
curve should be continuous with the line of the deposited cable, 
in order that this tension may be adequately sustained, we ought 
not to pay out the stray length to the extreme limit, but only 
until py becomes small ; for the prescribed movement of any por- 
tion of the cable, separately considered, must obviously require 
