of Action between, Magnets. 125 
VII., and VIII. have only to be multiplied by the error (in 
minutes) to give the error in H in terms of ly as unit. 
§ 11. Tables VI., VII., and VIII. give results only for two 
specified values of H and of m. Approximate values are, 
however, easily deducible for other values of H and m from 
the following considerations: or any given combination 
of distances, the relation between the error in wu and the 
consequent error in H is of the form 
COS 1 
0oH=/f(r)du. 
<tr a 
where /() depends only on the distances. Also, as a first 
approximation, we have 
| sin = 2m) Hr ; 
and so in the above equation 
OH =d777(7)Ouy.cos uw). HP /a.s ) wey't) ... (27) 
Now, in practice, the departure of cos from unity is 
seldom great. For u=20° we have cosu=0°94, and except 
at the shortest distance of deflexion, angles as large as 20° 
are unusual, Thus, treating du as given, we may accept 
SH « H2/m 
as giving a good approximation at all the larger distances of 
deflexion, and a fair approximation even at the least distance, 
unless H is exceptionally low. 
According to this approximation, the corrections to H in 
Tables VI., VII., and VIII., for a given value of H, when 
m= 1000 and when m=750 should be in the proportion 3:4; 
and the corrections to H, for a given value of m, should be 
four times as great when H=°370.as when H='185. Atthe 
larger distances in the Tables, it will be seen that these 
relations are very nearly satisfied indeed. 
§ 12. When one or more of the higher constants Q, R is 
not really negligible, the value found for P in the usual way 
from observations at two distances applies in reality to a 
different quantity, which we may call P’. As to the relation 
between P and P’, let us suppose that observations are taken 
at the two distances 7,, rz, and that P, Q, R are the only 
constants requiring to be taken into account. 
We clearly have from (1) 
1+ Py? =1 + Pr 7? 4+ Qry rt + Rr 7%, 
1+ Pro? =1+ Pr.-? + Qr.—* + Rr, 8; 
whence ee | 
Pl=P+ OCs Pr) + R(r,~4 Tite = teal eo) 
