72 
ASTRONOMY: A. O. LEUSCHNER 
At first Professor Newcomb was greatly surprised by the considerable dif- 
ference between our coefficients of the perturbations and those of Eichelberger, 
but formulae were soon derived which enabled me to deduce one from the 
other on the basis of the difference in elements used and thus to verify both 
sets of developments. After this investigation Eichelberger's original tables 
were published in an abridged form. 
More serious complications were encountered in the revision of other com- 
putations, which were due to misconceptions of theory and numerical errors. 
The integration of the differential equations for the perturbations gives 
rise to certain constants in the developments, to terms multiplied by the time 
(secular terms), and to purely trigonometrical or periodic terms. All of these 
added to the undisturbed coordinates give the disturbed place. Some of 
these terms are of the same form as terms employed in computing the undis- 
turbed positions and may be combined with them or introduced in the ele- 
ments. It then becomes merely a matter of arbitrary classification or com- 
bination, as to what part of the disturbed coordinate shall be directly com- 
puted from properly changed elements and what part shall be left over as 
perturbations. When all but the purely periodic terms are put into the 
elements, we may deduce a sort of mean elements, and we can thus speak 
also of a mean mean motion. These elements involve properly chosen con- 
stants and are not of the sort that they represent the actual position and 
velocity at any particular time. It was found that some of my predecessors 
in the work had misinterpreted the terms they were dealing with although 
their developments were numerically correct. 
The tracing of these and other inaccuracies for the planets to be revised 
have absorbed more time and effort than all the work on the planets that had 
not been previously undertaken. To simplify the determination of the con- 
stants and to secure greater accuracy Professor Newcomb 's program orig- 
inally called for adopting the mean of the several sets of elements printed for 
each planet in the B. J. as approximate mean elements, but later we shall 
see that in the case of Andromache the actual mean elements lie outside of 
the osculating elements obtained from observations and special perturbations 
over a period of thirty-three years. Thus we actually at first started with 
elements which were neither osculating nor mean elements, which involved 
corresponding difficulties later. But it should not be forgotten that at that 
time the whole problem of proper attack was still in an experimental stage. 
With reference to Brendel's very exhaustive and brilHant methods I may 
remark that I have come to the conclusion that even if they stand the test of 
the very critical cases of near commensurability, for practical purposes they 
do not have the advantage of the same simplicity as a proper choice between 
Hansen's and Bohhn's methods. 
For six of the twelve sets of tables published in 1910 comparison of 
observations taken in 1912-13 have been made with theory by Miss A. R. 
Kidder, with an agreement of less than the proposed 1' of arc, excepting in one 
