IN WOOD-BUILT AND IRON-BUILT SHIPS. 
59 
thus taken from the Table be called A^, and the correction required be called A^. 
Then A'-}- A" ; and 
A/^= ( A' — AJ -b A"=/:A(sin A' + a — sin A" + a) + A" nearly 
=(jij{sm A"-ba+A" — sin A" + a) + A"= A"( 1 +(j^ . cos A" +a) nearly. 
If in the computation of this small quantity we reject powers of jM/ following the first. 
A^^=(l— cos A"-ba).P. (sin 2A+|«/.sin A — a). 
But sin2A=sin2A"-b2jM-.cos2A".sin A'+ci nearly. 
After all reductions, A^^=P.sin2A"+/AP.cos 2A".sin A"H-a. 
The first term of this expression is in the very convenient form of a quadrantal 
term referred to the apparent azimuth of the ship’s head. The general influence 
of the second term is, that it produces no elfect on the maxima of the quadrantal 
terms, that it slightly increases the polar-magnet-deviation when A"=0 or 180°, and 
slightly diminishes it when A"=90° or 270°; and this will be practically a sufficient 
description of its characteristic effects. 
But as, in the aggregate of numbers, small terms become sensible which are scarcely 
sensible in the individual numbers, it will be desirable to ascertain the effect of this 
on a semicircular group. Combine the term //;P.cos 2A".sin A"-ba with the approxi- 
mate polar-magnet-deviationjo/.sin A"-ba, and integrate from A"-b«=0 to A"-ba=^: 
the result is 2|a-^l — ^ cos 2 ci^. Without the small term we should have obtained 
Hence it appears that the result for modulus found from semicircular groups, which 
p 
may be called the “Approximate Modulus,” must be multiplied by I + 3 cos 2a in 
order to obtain the “True Modulus.” Again, conceive the “approximate starboard 
angle made by the compound polar-magnet-force with the ship’s keel” to be a-bjS; 
then a semicircular sum from A"+a-bj8=| to ought to vanish ; the 
. . . r 2P I 
integral between these limits, omitting P/3 and (3^, is 2|sin (B — ^ sin 2a| ; hence 
2P . 
(^=~§~ sin 2a ; and this quantity must be subtracted from the “ Approximate Starboard 
Angle,” or a-bjS, in order to obtain the “True Starboard Angle,” or a. 
Thus it appears that, in the column of Tabular Polar-Magnet-Deviations, we are 
not comparing the tabular deviation due to the modulus and the starboard angle a, 
( P \ 2P 
1 — 2 cos 2a j and starboard angle a-f — sin 2a; and 
therefore our residual numbers ought to represent 
^.sinA"-|-a-l-P.sin2A"-l-jM<P.cos2A".sinA"-l-a— — ^cos2aysin^A"-l-a-l-^sin2a^. 
