CHARACTER OF THE ARMOUR-PLATED SHIPS OF THE ROYAL NAVY. 315 
Sphere. 
Let the centre of the sphere be at a distance r from the centre of the compass, and 
let r make angles a, (3, y with the coordinate axes to head, to starboard, and to nadir, 
and let 
4t r 
3 3 
P S - 
An 
M. 
Then 
a=M(S cos 2 a — 1), 
b=d = M 3 cos a cos/3, 
whence 
c=g = M 3 cos a cosy, 
e=M(3 cos 2 /3— 1), 
f=h = M 3 cos (3 cos y, 
&=M(3cos 2 y— 1), 
A = 
1+y {1 — 3 cos 2 y}, 
21= 
o, 
95 = 
M 0 , . 
— o cos a cos y tan 0, 
(5= 
M 
— 3 cos (3 cos y tan 6. 
2>= 
y • (cos 2 a — cos 2 /3), 
<$ = 
— 3 cos a cos p, 
From these we see that a sphere, wherever placed, will increase X and give a — k if 
1 
cos y<— 
' V 3 
or 
y>54° 45', 
and will decrease X and give a -\-k if y<54° 45'. 
Hence if, as before, we suppose a double cone traced out by a line passing through 
the compass, making an angle 54° 45' with the vertical, all spherical masses of iron 
whose centres are placed without the cone will increase the directive force and diminish 
the usual heeling error. All spherical masses whose centres are placed within the cone 
will diminish the directive force and increase the heeling error. Hence, as far as pos- 
sible, no iron should be either below or above the compass within an angle of 54° 45' of 
the vertical passing through the compass. 
If cos a > cos (3 , or if the centre of the sphere be in either fore-and-aft quadrant, the 
