Mr. Woolhouse on the Deposit of Submarine Cables. 351 
not differ much from the limiting angle X; and as a small error 
in @ would then considerably affect the other values, it will be 
preferable to determine this angle by calculation. For this pur- 
pose we have 
oui s _9 Sin (A—@) tan ir coat 
“tee aiges - sin A tan 1 (A—o) (k4) 
If for given values of é or X, values of this expression be caleu- 
lated and tabulated under o, then by entering this Table with 
the known values of 7 the angles @ will be readily deduced. 
From (12); 
—2e(s—2x) 
tan? ig— YO ; 
5 ¥+2po 
; _ y= 2e(s—z) _ 
Po tan?i@ : 
Hence we conclude that when, by extra paying out, the amount 
of “slack” or “stray length” (s—a#) is increased, and the in- 
clination @ also increased, the radius (pg) of curvature at the 
lowest point of the curve becomes sensibly diminished ; and it 
will be evanescent when 
Qe-—* = 1—tan*do. 
y 
The value of the constant p, will perhaps be best determined by 
the formula 
ppg el ed, vie cath; thf) 
and the amount of stray length by the formula 
PP var (T—a) cos a— py 
e 
y —(T—a)(1—cos oa) 
2 cotr 7 
which are deduced from (11) and (12). For the calculation of 
these the observed value of » will probably be sufficiently 
accurate. 
We have only further to remark that the curve has a recti- 
linear asymptote inclined at the limiting angle » with the hori- 
zontal; that the horizontal distance of this asymptote from the 
lowest point = Pa, and the horizontal distance from any other 
(16) 
point of the curve = this distance multiplied by z. Also if 
from any point in the curve a tangent be drawn terminating in 
