6 
ASTRONOMY: SEARES, VAN MAANEN AND ELLERMAN 
with the numerical solution and the necessity of limiting what at best could 
be only a very laborious undertaking, the observations were confined to the 
zone 45°N-45°S. Twelve spectrograms with exposures of 10 to 30 minutes 
constituted the normal observing program for each day. For 63 of the days 
the photographs have been completely measured by van Maanen, who has 
assumed the responsibility for this part of the undertaking. More than 2000 
sets of measures were required, each involving about a hundred settings of 
the micrometer. 
In measures of minute displacements of spectral lines, systematic errors 
are always to be suspected, as well as the influence of prejudice arising from 
a knowledge of the results that will satisfy a given hypothesis. Such syste- 
matic errors as may have entered in the present case probably affect only the 
constant k, which varies from line to line but does not enter into the determi- 
nation of the position of the magnetic axis. 
To exclude the influence of prejudice, the procedure devised by Mr. Hale 
has been followed here. The limited zone of heliographic latitude covered by a 
single spectrogram may lie in the northern hemisphere, where the displace- 
ments are positive, or in the southern hemisphere, where they are negative; 
or it may extend over the equator and thus show only very small displace- 
ments, some negative and some positive. The measurer has rarely known in 
advance the latitudes covered by any spectrogram. Further, the above dis- 
tribution of algebraic signs presupposes that the photograph has been made 
with the compound quarter-wave plate in its normal position. Since the in- 
version of the plate reverses the signs of the displacements, its position, as a 
final precaution, has been varied at random by the observer, and the measurer 
has not known the position used for a given photograph until after his settings 
were finished. 
The data have been treated as follows : Each displacement affords an equa- 
tion of condition of the form (1) for the determination of the unknowns k, i, 
and X. The longitude of the magnetic pole, X, involves an epoch, to, when the 
pole was on the central meridian, and the period, P, in which the magnetic 
axis revolves around the axis of rotation. For a single day we may assume 
X to be constant, which leads us to discuss separately the observations for 
each day and for each line, thus deriving values for two new unknowns, x and 
y, which are functions of k, i, and X. The analysis of x and y for the whole 
series of days then determines k, i, P, and to. 
Equation (1) may be written 
Ax + By = A. (4) 
A and B are the bracketed expressions of (1), including only known quantities, 
and 
x — k~ l cos i, y = k~ l sin i cos X (5) 
whence 
Y = y/x = tan i cos X. (6) 
