CHARACTER OF THE ARMOUR-PLATED SHIPS OF THE ROYAL NAVY. 269 
*=l+“-±£, Sf=±A, 
®= «=*£* e=‘(/tan.+g). 
From equations (7) and (8) we obtain 
, v 31 + 33 sin £ + (£ cos £ + ® sin 2£ + (5 cos 2£ SC)\ 
an 1 + 33 cos £— CE sin £ + 2) cos 2£— (S sin 2%’ 
whence if £' be the azimuth of the ship’s head measured from the direction of the dis- 
turbed needle so that %'=% — 
sin &=9l cosd+93 sin £'+($: cos£'+T> sin (2£' +&)+($; cos(2£'-|-&). . . (10) 
If the deviations are small, we have approximately 
S=A+B sin£'+C cos£'+D sin2£'+E cos2£', (11) 
in which A, B, C, D, E are (nearly) the arcs of which 9t, 93, (5, 2), (S are the sines. 
The term 93 sin £' + (S cos £' may be put under the form \/93 2 +(£ 2 sin (£'- \-a ), in which 
a, called the starboard angle, is an auxiliary angle such that tan a=|^ • 
If the soft iron of the ship be symmetrically arranged on each side of the fore-and-aft 
line of the ship through the compass, then 
4=0, d= 0, /= 0, 
91=0, <g=0, 
A=0, E = 0. 
R • 
If we put |«,=l+^+^5 the expression of the nadir force of earth and ship in terms 
z 
of earth’s vertical force as unit, is 
Nadir force = |= t -£ 1 cos?-^sin?+ f ., (12) 
If the ship heels over to starboard an angle i, 93 and 5) ( 0 r B and D) remain unaltered ; 
and representing the altered values of 91, (5 and Cs by 9t f , (5 t , and ($ t , we have 
a ‘= 3 t -V*’ 
s.=G- ($ + £-l) tan i 
The alteration in 9t and (£ may generally be neglected ; that in 6 is often of great 
importance. The quantity %= 1^ tan 6 is called the heeling coefficient, and 
represents the degrees of deviation to windward, or the high side of the ship, produced 
by a heel of one degree when the ship’s head is North or South by the disturbed compass. 
