THE COMPASS-NEEDLE ON THE DEVIATIONS OF THE COMPASS. 
175 
on each other is not difficult. Such needles, if otherwise acted on by the same forces, 
will remain parallel to each other. Let AB A'B' represent the needles, a their common 
length, 5 their mutual distance, and m their common force. 
The force of each to turn the other will be 
aW- 
and if H be the horizontal force of the earth, they will produce a deviation 
mb y.r 1 11 
3 m a 
2 H ^ 
24 
If one of these needles be directed towards the other, and the binnacle be moved in 
azimuth till the needles are at right angles to each other, the deviation produced in the 
second needle will be 
a f- 1 
The quadrantal coefficient, or the greatest deviation which the needles will produce 
on each other when left free, is therefore three-fourths of the greatest deviation which 
one of the needles can cause in the other when placed in the most favourable position. 
2. The proportion of the octantal term introduced to the quadrantal is 
24 
^+ 4^2 
2 * 
If ^ = i this term is = = — , so that with a quadrantal deviation of 6 ° 30' this 
b 3 226 6-5’ ^ 
should introduce an octantal de\iation of 1°. We may therefore safely fix three times 
the length of the needle as the limit of distance within which single-needle compasses 
should not be allowed to approach each other in order to avoid the octantal error. It 
will be observed that this theoretical result coincides, as nearly as possible, with the 
experiment which gave for a quadrantal deviation of 6 ° 41', produced by two needles at 
a distance of three times their length, an octantal deviation of 0° 58'. 
3. If instead of two single-needle compasses we have the reciprocal action of two 
double-needle compasses, the distances of the ends from the diameter which is parallel 
to the needles being a, the deviation produced is 
-3 
ma cos a 
{ ^ 32 cos" Ct) sin 2 ^'— 
35 cos 3a 
24 b^ cos a 
Sin 
SO that the octantal term is made to vanish by the same arrangement of two needles 
that we have already described. 
