244 
ASTRONOMY: SEARES, VAN MAANEN AND ELLERMAN 
cal values of the coefficients A and B. There is no change in algebraic sign 
and the range in their values is so small that the simultaneous determination 
of X and y becomes practically indeterminate. For the middle zone the condi- 
tions are more favorable, but even here it has not seemed advisable to at- 
tempt a direct solution. Since the quantitj^ as shown by the uniform-field 
solution, is apparently constant throughout the entire interval, we have pre- 
ferred to solve by successive approximations, beginning with an assumed value 
of X, 
We therefore write the fundamental equation in the form 
B tan i cos X = ^x' — A, (3) 
where for convenience x' = ^cos i has been substituted for 1/x. The value 
of x' from the uniform-field solution is 1.00, and this we use for the first ap- 
proximation in deriving Y = tan i cos X from (3). We have for each zone, 
from all the data for each day, the normal equation 
[BB] tan i cos X = [BA] x' - [AB] (4) 
The individual values of Y are presumably in error, because of the assumed 
value for x\ It is easily shown, however, that neither the phase nor the 
period of the curve Y = tan i cos X is thereby affected. The entire effect goes 
into the amplitude. Using the first approximation for x', we can therefore 
determine to and P for each zone, free from error, exactly as in the case of the 
uniform-field solution. The curves for Y show that the results in each case 
are sensibly the same as those previously found, and we therefore adopt for 
each zone the values given above. 
Since the weight of tan i cos X given by (4) is [BB] the normal equation for 
tan i is 
tan i S [BB] cos^ \ = x' X [BA] cos X - S [AB] cos X (5) 
where the outer summation symbol covers all the separate days. With this 
equation we find for the three values of i 
I 4?9 ± 0?7; II 6?7 ± 0?5; III 3?7 ± 1?0 (6) 
With the aid of these values we now determine from the measures of each 
day a new approximation for x\ using the normal equation 
[AA] x' = [AA] + [BA] tan i cos X (7) 
which is at once derived from the fundamental equation. Noting that the 
weight of each x' is [AA] and combining the different days, we have for the 
mean 
x'X[AA] = X[AA] + tan i X[BA] cos X (8) 
The results of the second aproximation for x' are 
I 0.97 ± 0.009; II 0.60 ±0.018; III 0.96 ± 0.008 (9) 
