170 • MESSES. SMITH AHD EVANS ON THE EEEECT OE THE LENGTH OE 
Let — m' be the force of the other end B' of the needle, for which a— ~a. 
The force of — m' to turn B' round C is therefore 
and the whole force to turn the needle round C, 
I ® f I ^ • w I 2 ® m—m' . 15 „ ,1 
=M(»«+»«')p|(l+8 Fj 6 ^ psm 3?'|. 
It appears from this equation that the eifect of one end of a bar magnet on a single- 
needle compass of finite length is, — 
1st, to increase the deviation which would be produced in an infinitesimal needle in 
3 a® 
the proportion of 1+g ^ • 1 j 
15 
2nd, to cause a sextantal deviation of which the coefficient is or five times the 
increase of the semicu’cular deviation. 
• 3 ^ Tit TTiJ 
3rd. To cause a quadrantal deviation of which the coefficient is 2 ' ^ * m+m' ' 
If the two ends of the needle are of equal strength, this term disappears ; but if one 
end is stronger than the other, there will be a small quadrantal deviation, the effect of 
which will be shown in the diagram by a shifting of the points of intersection of the 
long needle curves with the short needle curves. 
When m=m', the force to turn the needle 2 a in length about its centre is 
2Mmf,|(n-| |sin3c} (1.) 
It will easily be found from these formulse that the force of one end of a magnetic 
bar to turn two parallel needles placed symmetrically on each side of a diameter, the 
ends of each being a from the end of the diameter, is 
= 4Mw ^ cos a| (^1+1 sin 
15 cos 3« 
8 b'^ cos « 
( 2 .) 
in which a represents the diameter of the circle of which the needles are respectively 
chords. The coefficient of the sextantal deviation, having cos 3a for a factor, will be 
zero if 3a=90° or a=30°. Hence, if the two needles are placed with their ends 30° on 
each side of the diameter, and therefore 60° from each other, there will be no sextantal 
deviation. This is the simplest form of the Admiralty Standard compass. 
If there are two pairs of parallel needles placed symmetrically on each side of a dia- 
meter, the ends of the needles being distant a and a' from the extremities of the diameter, 
the force to turn the card is 
a « — cJ 
■ 4Mm To cos - 7 r~ cos — 
b‘‘ 2 2 
cos 3 — - — cos 3 
I ^ I 1 5 2 2 ' oyi / Q ^ 
P+8 • • (3-) 
COS r COS 
