Mr. Woolhouse on the Deposit of Submarine Cables. 358 
From the latter of these equations, 
T—a=p (cosw—e*sin?w); . . . © (8) 
and at the lowest point where »=0, Tj—a=ppo. Equations (2) 
also give by division, 
: B! (2 io 
aT ie sin @— 3 ( —-—cos@ : 
T-—a_ cos @—e* sin? w MG hc al 
To facilitate the integration of this expression, let X be the 
limiting angle of the curve or value of w which makes 
cos w—e* sin? w=0. 
Then 
O= cos \X—e? sin? A=e? cos? 1+ cosA—e?; 
and if for brevity we put cos A=a, « and — : will be the roots 
a 
of the quadratic, 
Therefore 
1 } 
ees eee ee oe a a Eh 
cos @—e? sin?w 
e? (cos wo—a)(cos o+-) 
a 
iat mae 1 1 ) 
~ 1+a?\coso—a 1)" 
+a sS@—a o+= i 
By substitution and integration, 
1 
Tg Lan? cOs@ = 
c Tide a2 eee 
m 
BI P\ (Face) cos o) oo — cos o) 
- B Tea ve cos@—a 
log 
oe Fite Ty ee 
cos@+ — 
a 
tegrating, the accumulated force, as measured by weight in water, =—-V. 
Therefore the vertical effective force Bie sin A,~and the horizontal force 
=——% (1—cosA). If the expressions (1) and (2), instead of being put 
equal to zero, be respectively equated with these, the only effect on er 
equations (3), (4) which result from them, will be the substitution of i” = 
for ¢ in the latter, and they are then identical with our equations (2). 
2 
The correction a or a, is always a very small quantity. 
