172 MESSRS. SMITH AND EVANS ON THE EFFECT OF THE LENGTH OF 
the above ratio becomes h : which represents the advantage gained in dimi- 
nishing the sextantal deviation by the correcting magnets being placed as nearly as 
possible below or above the compass. 
This expression, compared with equation (1.), shows that the same arrangement which 
prevents a sextantal deviation when the magnet is in the same horizontal plane does so 
when it is elevated or depressed. 
The expression for the semicircular deviation shows that it is increased or diminished 
by an increase of the length of the needle, according as h is less or greater than ^5- 
If we desire to know the amount of sextantal deviation produced by a bar magnet on 
a single needle in the same horizontal plane, we must consider the effect of both ends 
of the magnet. The expression becomes, however, too complicated for use when the 
problem is stated generally. In order to simplify the formulm, we may consider the 
magnet directed first endways and then sideways to the needle. 
1. If the bar magnet be in the same horizontal plane and directed to the needle: — 
Let h and V be the distances from the centre of the needle to the two ends of the 
magnet. 
The force to turn the single needle compass is 
|l+| (p+p)} sin ^ 3^' 
. . ( 6 .) 
If the magnet be short this becomes 
= 4Mf sin^' + f 
2. If the bar magnet be in the same horizontal plane and directed sideways. Let |3 
be the angle subtended by the half of the magnet. 
Force . . . . (7.) 
^ being the angle between the needle and a line parallel to the magnet. 
If in the last equation the correcting bar be short, /3 will be small and = 3, and 
proportion of the coetficient of the sextantal part to that of the semicircular part 
will be 
45 - 3 
"s Y • -‘-+8 F 
If 1=^5 or if the distance of the magnet be six times the length of the needle, this ratio 
will be 
25 - 7 ' 
If the semicircular deviation is very large, the sextantal deviation should 
not be allowed to exceed this proportion, and therefore with single bar needles the 
correcting compass should not be placed within six lengths of the needle of it. 
The sign of the sextantal part being the same as that of the semicircular in equation 
