130 Dr. C. Chree on the Law 
f(z), and so P(A, 2’), vanishes when -='8165. It is 
positive for smaller, 1and negative for all larger values 
of z; it diminishes algebraically continually as z increases 
from 0. 
F,(z), and so Q(A, A’), vanishes when -=°4667 or 1°565. 
It diminishes algebraically as z increases from 0 to 1°155, 
where there is an (algebraical) minimum. It is negative 
when < lies between *4667 and 1°565, otherwise it is positive. 
f(z), and so R(A,X’) vanishes when z=°3318, or °9048, 
or 2°252. It is positive when z lies between 0 and °3318, or 
between °9048 and 2°252. Outside these limits it is negative. 
There is an algebraical minimum (numerical maximum) 
when <='6714, an algebraical (and numerical) maximum 
when z=1'884. 
On the hypothesis NA=U/L, 
inferences can be drawn from Table XI. as to the best ratio 
between the lengths of the two magnets, assuming the 
terms depending on the cross section negligible to a first 
approximation. 
Where the horizontal force is very high, as in India, it 
might be best, from a purely theoretical standpoint, to secure 
that R is negligible by making /’/] equal :3318 or a slightly 
larger value. This has the advantage of making Q small 
as well as R. 
Where the force is comparatively low, and deflexion-angles 
are sufficiently large at distances so considerable that R may 
be left out of account, a good deal is to be said in favour of 
the ratio !/l="4667, 
as reducing @ to zero, and so rendering observations at more 
than two distances theoretically unnecessary. 
Where the force is so small that very large deflexion 
distances are necessary, the value 
U/l=-8165, 
for which P vanishes, has a good deal to commend it. 
In polar regions, where the horizontal force is low, 
magnetic disturbances seem to abound, and climatic conditions 
are unfavourable ; there are thus obvious advantages in an 
arrangement which admits theoretically of a high degree of 
accuracy from deflexions at a single distance. 
§18. Table XII. gives numerical results for representative 
English magnetometers, assuming 2A//l/ = 2d/1 = p, and 
assigning a series of values to p. The values of P’(A, 2) 
assume the two deflexion distances to be 30 and 40 cms. 
