Mr. Woolhouse on the Deposit of Submarine Cables. 359 
B’, has so little influence on the form of the curve that its effects 
may be practically disregarded, and their omission will greatly 
simplify the formule, The differential equation (4), from which 
we have obtained the formula (8) for determining the tension, 
obviously indicates, when m=n, that by neglecting the longitu- 
dinal friction the value of the tension will be sensibly increased 
only towards the upper extremity of the cable, and even there 
but slightly, as the coefficient B! is small as compared with B. 
Therefore by (8) the radii of curvature towards this extremity 
will also be slightly inereased in the same proportion. But as 
this portion of the curve is nearly straight, it is evident that a 
small proportionate augmentation of the large radii of curvature 
cannot produce any sensible divergence throughout the limited 
extent of the curve we have under consideration. For a given 
value of » the coordinates r= f p dw cosa, y=f pdwsine will 
be doth slightly increased ; but the poimt may be considered to 
be merely transferred along the curve, as the divergence towards 
the convex side will be extremely minute. We shall therefore 
deduce the formule assuming B!/=0, which will be more conve- 
nient, and sufficiently accurate for all practical purposes. 
Making B’/=0 in (6), we get 
—_ — 1—o2 22 eos 2 
T a l—a@ acoswt+] ea { tan? 3 } ‘| (10) 
tan (9—) 
Po l+a coswa—a 
Hence if T,, , denote values referring to way given point, we 
shall have 
T-a a(eeer seine Vise = { oot (0—p) a (11) 
T,-—a~ \acosa,+1 coso—a cot (0; —) 
Ci) A i. 
Again, since cos #— e* sin? a =e? (cos a— a) (cos a+), from 
(3) we get 
T—a __p cosw—a acoso+l | 
Ti—a ?p, cOs@,;—a acos@,+1’ 
cosa—a \__?_ f/acosw+1\_-2%. 
= ——_——_—_— 1+o2 
) (seer - + (12) 
1+a? 
Pi COS @,— a 
F P 
Also the first of the equations (2) gives, by integration, 
ME ee Rad teks saat Sides te cag sts LO) 
— 1—a 
y l—a acoswt+l reac (ys 
"po \l+a cosw—a 
which is the equation of the curve. Or, adopting the former 
constants, 
ee oo en 
Po tan (@—p) (5) 
. (14) 
2B2 
