ASTRONOMY: SEARES, VAN MAANEN AND ELLERMAN 245 
It thus appears that for zones I and III the original assumption was nearly 
correct; for the second zone, however, the change in x' is large. 
For the further approximations we have from (5) and (8), respectively, 
d(tSLn i) — adx' 
dx' = i(?f/(tan I) 
where 
S [^A] cos X _ S [^A] cos X 
S {BB\ cos'X 2 [AA] 
For zones I and III, a. and /8 are small quantities of the order of 0.01 or 0.02, 
and since dx' ^ the differences between the first and second approximations for 
x\ are only —0.03 and —0.04, respectively, equations (10) and (11) show that 
the corresponding values of i and x' given in (6) and (9) are final. 
For zone II, a — 0.120 and ^ = 1.42. We have found already, in order, 
x\ = 1.00, tan ii = 0.118, a;'2= 0.60, whence dx' = —0.40; and by successive 
substitution into (10) and (11) we derive tan Z2 = 0.070, ^c's = 0.53, tan iz = 
0.062, a:'4= 0.52, tan = 0.061. As no further change is produced by addi- 
tional substitutions, x'^ and tan are adopted as the final values. 
Calculating k from x' and collecting results, we have 
For ^ = 45° N to 10° N, i = 4?9, = 0.96 
= 10 N to 10 S, = 3.5, = 0.52 
= 10 S to 45 S, = 3.7, = 0.96 
P = 31.52 ± 0.28 days 
to = 1914, June 25.38 ± 0.42 days, G. M. T. 
The uncertainty in the inclinations averages about 0?7, while that in the ^'s 
is one or two units of the second decimal. Since the inclinations, periods, and 
epochs for the three zones are sensibly the same, the deviations from a uniform 
spherical field within the region from 45° N. to 45° S. are apparently restricted 
to the polar field-strength, which is inversely proportional to k. Here the 
variation with latitude is unexpectedly large — far greater than the uncertainty 
of the calculation. Unless the observations are affected by a large systematic 
error, which is itself a rather complicated function of the magnitude of the 
observed displacement A, the change must be real. 
The measures have all been made with the parallel-plate micrometer and 
it is difficult to believe that errors of the kind required to make the result il- 
lusory can have entered. Our experience would lead us to believe that im- 
portant variations in the field-strength of the character indicated actually 
exist. 
It is unexpected to find the i's for all the zones smaller than that from the 
uniform-field solution. It is easily seen, however, that the two series of re- 
sults stand in the proper relation to each other. In the uniform-field solution 
the displacements within the limits of zones I and III contribute about fifteen 
(10) 
(11) 
(12) 
