applied to Terrestrial Magnetism. 458 
The quantity v in formula (2) is the sum (or difference) of the moments of the 
weight of the needle and of the added weight. Accordingly, if m denote the mass 
of the added weight, and a the distance of its point of application from the axle, 
v-ph= mag; 
or, since n=p v, 
Now the force of gravity, g, varies with the latitude ; and the variation is expressed 
by the formula 
£=g (l1—e cos 2), 
in which g" is the force at the latitude of 45°, \ the latitude of the place of ohsery- 
ation, and e a constant whose numerical value is .002588. Accordingly, substituting 
this value of g in the preceding expression, and employing the symbol v’ to denote 
-, mag 
the value of » corresponding to the latitude of 45°, or the quantity gyre have 
v=v 1—e cos 2X). (6) 
The equations (2) (5) (6) contain all that is requisite for the comparison of the 
magnetic force at different places of the earth’s surface, and under different circum- 
stances as to temperatnre. The expression for the force, obtained from them by sub- 
stitution, is 
v' cos 8 1—e cos 2d 
ia o’sin(d—8@) * feweor) J 
(7) 
This expression is peculiarly adapted to logarithmic computation : for, since e cos 2d 
and a (r—r) are very small fractions whose squares and higher powers may be ne- 
glected, 
log (l—e cos 21) = — Mecos2d, 
log[1—a (r—7') ] = — Ma (7-7); 
M being the modulus of the common system, whose numerical yalue is .43429. Hence, 
if we take 
A = log cos 6 — log sin (8—0@) + Ma (+—7') — Me cos 2X, 
4 = log cos 0, — log sin(8,—6,) + Ma(z,—7) — Me cos 2X,, (8) 
in which 8, 6, &c. denote the values of 8, 0, &c. at the place for which the force is 
taken as unit, we have 
log ¢= log v’—log o' + A, 
O= log v —loga' + A; 
