September 18^ 1914] 



SCIENCE 



393 



for that angle on the sum of the numbers 

 extracted from the tables. 



The problem of fitting the terms into the 

 smallest number of tables is a problem in 

 combinations — something like a mixture of 

 a game at chess and a picture-puzzle, but 

 unlike the latter in the fact that the inten- 

 tion is to produce ease and simplicity in- 

 stead of difficulty. This work of arrange- 

 ment is now completed and, in fact, about 

 five sixths of the calculations necessary to 

 form the tables are done ; over one third of 

 the copy is ready for the printer, but, 

 owing to the large mass of the matter, it 

 will take from two to three years to put 

 it through the press. The cost of perform- 

 ing the calculations and printing the work 

 has been met from a fund specially set 

 aside for the purpose by Yale University. 



A few statistics will perhaps give an idea 

 of our work. Hansen has 300 terms in his 

 three coordinates, and these are so grouped 

 that about a hundred tables are used in 

 finding a complete place of the moon. "We 

 have included over 1,000 terms in about 

 120 tables, so that there are on the average 

 about eight terms per table. In one of our 

 tables we have been able to include no less 

 than forty terms. Each table is made as ex- 

 tensive as possible in order that the interpo- 

 lations — the bane of all such calculations — 

 shall be easy. The great majority of them 

 involve multiplications by numbers less 

 than 100. There are less than ten tables 

 which will involve multiplications by num- 

 bers between 100 and 1,000 and none 

 greater than the latter number. The com- 

 puter who is set to work to find the longi- 

 tude, latitude and parallax of the moon will 

 not need a table of logarithms from the 

 beginning to the end of his work. The rea- 

 son for this is that all multiplications by 

 three figures or less can be done by Crelle 's 

 well-known tables or by a computing ma- 

 chine. But Mr. Hedrick has devised a table 



for interpolation to three places which is 

 more rapid and easy than either of these 

 aids. It is, of course, of use generally for 

 all such calculations, and arrangements 

 are now being made for the preparation 

 and publication of his tables. The actual 

 work of finding the place of the moon 

 from the new lunar tables will, I believe, 

 not take more time — perhaps less — ^than 

 from Hansen's tables, as soon as the com* 

 puter has made himself familiar with them. 

 Fortunately for him, it is not necessary to 

 understand the details of their construc- 

 tion : he need only know the rules for using 

 them. 



1 am now going to show by means of 

 some diagrams the deviations of the moon 

 from its theoretical orbit, in which, of 

 course, errors of observation are included. 

 The first two slides exhibit the average 

 deviation of the moon from its computed 

 place for the past century and a half in 

 longitude.^ The averages are taken over 

 periods of 414 days and each point of the 

 continuous line shows one such average. 

 The dots are the results obtained by New- 

 comb from oecultations ; the averages for 

 the first century are taken over periods of 

 several years, and in the last sixty years 

 over every year. In both cases the same 

 theory and the same values of the constants 

 have been used. Only one empirical term 

 has been taken out— the long-period fluc- 

 tuation found by Newcomb having a period 

 of 270 years and a coefficient of 13". I 

 shall show the deviations with this term 

 included, in a moment. 



The first point to which attention should 

 be drawn is the agreement of the results 

 deduced from the Greenwich meridian ob- 

 servations and those deduced from occulta- 

 tions gathered from observatories all over 

 the world. There can be no doubt that the 

 fiuctuations are real and not due to errors 



2 Monthly Notices S.A.S., Vol. 73, plate 22. 



