of the Earth’s magnetic Force in absolute Measure. 5 
in which @ and q@’ are unknown coefficients, the former of which is double of the 
ratio sought. Accordingly, the method recommended by Gauss consists in ob- 
serving the angles of deflection, w and w’, at two different distances, p and pb‘, and 
inferring the coefficient @ by elimination between the two resulting equations of 
condition. 
The object of the present paper is to point out the means by which the 
quantity sought may be obtained, without elimination, from the results of obser- 
vation at one distance only; and thus not only the labour of observation be 
diminished, but (which is of more importance) the accuracy of the result in- 
creased. Before entering on this, however, it will be expedient to ascertain the 
amount of the probable error in the received method. 
The coefficient of the first term, obtained by elimination between the two 
equations of condition above alluded to, is 
p”® tan wu’ — p’* tan u 
2 
Q=— 73 
Dee 
The distances being greater than four times the length of the magnets, the 
angles of deflection are small, and there is, approximately, tan w = wu tan 1’, 
tan uw’ = w' tan 1’, wand’ being expressed in minutes; and making pb’ = gp, the 
preceding expression becomes 
sean Vl —U 
Q= pv’ tan 1’ ——_—_—_. 
¢g—1 
The probable errors of uv and w’ are equal ; and, by a well known theorem of the 
calculus of probabilities, the probable error of Q is 
V q+ 
Te 
In determining the ratio of this error to the quantity itself, we may observe that 
there is, approximately, g°w’ = wu, and 
Aq = p* tan 1’ Au. 
Q =p? tan ee; 
and dividing the formula last found by this, 
Ae _ gh Fl au 
Qu treage eee 
It appears from the preceding theorems, that the value of Ag, corresponding 
to a given value of Aw, varies with the assumed ratio of the distances, 7; and 
