174 
IMESSES. SMITH AND EVANS ON THE EEEECT OE THE LENOTH OE 
. u + a! , a — a' 
COS 5 — -r — COS5-— r — 
If, as in the Admiralty compass, —^=15°, the factor 
cos 
a + a' 
becomes 
cos 
2 2 
—tan 15°,. or the decantal term is reduced to about one-fourth of the amount produced 
by a single needle. 
Expression (6) becomes 
1 ^ 
Ml 
1 i-f- g « ^ 
j2-i- in ^ -1-04 » \i 
b-)j 
+ 
15 
' (^+^) ^ + p) 3?'+ a‘ (|4+s 45+^) sin 5^' 
If the magnet be short this becomes 
=4Mm« -^1(1+4 F+M Jjj + (-4 f+T^8 ¥) 
Expression (7) becomes 
^ ' o \ \ ^ I ■ v! I «^\sin3/3' . 
2M“F®m^|(.l+8 F+64 ¥) P+m 
, 315 af^ sin 5/3 . 
which, if j3 be small, becomes 
3 15 «4 
oi\r ® • /al I ^ I • yt i • ow i • kvi] 
2Mmpsm/3|(^l + g js+gj p) sin 3^'+^ ^ sin5?'|. 
Hence if the deviation produced by two short needles at equal distances, placed side- 
ways on the east and west side of the needle, be corrected by one short magnet of the 
same kind placed on the north side and directed endways, the residual error ivill be 
'S . 15 a\ . . /75 . 315 aS . 315 a'* 
— I «Tn 
64 
{(I p+M p) *“?'+' 
Dev. \ ( g p+g pj sin;'+(^ ^j) sin3^'-^ ^sinS^' 
so that by such an arrangement the sextantal and decantal errors may be obtained 
almost freed from the semicircular error, and the effect of different arrangements of 
needles in diminishing these errors more easily tested. 
These results, it will be observed, suppose the magnetism of the needles to be collected 
in one point at each extremity ; but as the points in which the magnetism may with least 
error be considered as collected lie a short distance from the extremities, and therefore 
must be considered as lying at greater angular distances from the central line than the 
extremities, the consequence is that in the Admiralty Standard compass cards the 
sextantal and octantal errors are slightly over-corrected ; and the accuracy of the correc- 
tion might doubtless be increased, with little or no injury to the performance of the 
compass in other respects, by bringing the needles a little within the regulated distances 
of 30° and 60°. 
Mathematical Inmstigation of the Effect of Two Needles on each other. 
1. The investigation of the deviation produced by two needles of equal size and power 
