THE COMPASS-NEEDLE ON THE DEVIATIONS OF THE COMPASS. 
173 
(6.), shows that its effect is to lower and broaden the curve ; the sign being different in 
equation (7.) when /3 is less than 30°, shows that its effect is to heighten and narrow it. 
The effect of soft iron correctors on the deviation of the compass, leaving out of con- 
sideration for the present the effect on the needle of the magnetism induced by the 
needle in the soft iron reacting on the needle, will be found by substituting for the con- 
stant coefficient M a coefficient M' sin cos which will represent the magnetic 
state of the soft iron having magnetism induced in it by the action of the horizontal 
force of the earth in the different positions of the ship’s head. 
It will be easily seen, without repeating the calculation, that this will give a constant, 
a quadrantal, and an octantal deviation instead of a semicircular and a sextantal devia- 
tion, as in the case of a bar magnet ; and that the octantal de\iation will be reduced 
to zero by the same arrangement of needles in the compound card which reduces the 
sextantal de\iation to zero. 
WTien one end of the needle induces magnetism in soft iron in its neighbourhood, the 
disturbing force is itself proportional to the force of the needle ; so that in the above 
equations we must substitute for M, so that the force to turn B round C will be 
sin ABC 
,BC.AC 
AB^ 
sin ACB 
=Xm^^{(l+^ sin^'+l ^sin2^'-f-^ • sin 3^'j. 
The comparison of this formula with those given above will show that the same con- 
struction of the compass which prevents a sextantal error arising from the length of the 
needle when a permanent magnet affects it, prevents the like error when the needle acts 
on and is reacted on by soft iron. 
we 
cany the expansion to the fourth power of expression (1) becomes 
2Mm ^ 
iression (3) becomes 
T , 3 15 _ /|5 ^2 105 , 315 . -y,! 
^ +8 +1^ F /• 
ecomes 
^2 COS 
« + «' a — a'r/ 3 15 fl‘*\ . 
2 ~| + 8 62 “ 1 “ 64 b'^ ) ^ 
3^'+ 
cos 
„ a + — m! ^oi-\-ot! ^ a. — 
+U P+128 ¥) — ™ 3^ 
COS — - — cos — - — 
128 S'* 
— — , sinor 
« + «' « — « ^ 
cos— ^ cos— ^ 
if 
a — d 
:18° 
’, or if one pair of needles be 12° from the diameter, the other 48°, 
involving sin 5^', which may be called the decantal term, will also vanish. 
