MADE BY CAPTAIN BACK DURING HIS LATE ARCTIC EXPEDITION. 
379 
w g cos (fi + y) — l m sin (fi — 
w g cos (0 — y) — l m sin (0 — 
8) = 0 j 
l) = ()J 
for the determination of the dip, without inverting- the poles of the needle, when the 
position of the centre of gravity of the needle is known. These equations, when the 
dip is known, determine the position of the centre of gravity. The form, however, 
in which I have given the expression for the determination of the dip, 
tan l = 
2 — (cot '9 — cot fi) cot y 
cot '9 + cot fi 
is not so convenient for computation as another which it may be made to assume. 
The above equations may be put in the form, 
M sin ('0 — o) ■— cos (0 — y) = 0 
(!•), 
M sin (fi — l) — cos ( t 6 + y) = 0 (2.) ; 
where M will represent the ratio of the static momentum of the magnetic force acting 
upon the needle, to the static momentum of its weight, about the axis of motion ; 
& representing the inclination of the direction of the terrestrial magnetic force to the 
horizon, or the dip ; y the angle which the line joining the centres of gravity and 
motion makes with the magnetic axis of the needle ; '0 and fi the angles which that 
axis makes with the horizon, when the centre of gravity is above the axis of motion, 
and when it is below that axis. From these equations we obtain 
M {sin (0 — c5) + sin ( t 0 — 1)} — {cos (0 — y) + cos ifi + y)} = 0, 
M {sin (0 — c>) — sin ( t 0 — £)} — {cos ('0 — y) — cos (,0 + y)} =0. 
Consequently, 
M . cos 
i 
M . sin 
'9 - ,< 3 
O 
o 
^ - * 
^ - a 
} 
} 
'9 + fi 
— COS cos 
'9 + yfl . 
— sin — ~ sin 
/g _|_ (j IQ Q 
or, putting S = and D = — ^- L , 
M cos D sin (S — S) — cos S cos (y — D) = 0 (3.), 
M sin D cos (S — l) — sin S sin (y — D) = 0 (4.). 
Hence we have 
tan S . tan (S — S) = tan D . cot (y — D) (5.) ; 
a most convenient equation for computing the value of £, that of y being known ; or, 
vice versa, for determining the value of y from that of &, to which purpose I shall now 
apply it, with reference to the observation made at Fort Reliance with the needle 
No. II. 
In the following Table are given the means of five observations made with the 
* Philosophical Transactions, 1833, p. 345. 
3 c 2 
