22 Report S.A.A. Advancement of Science. 



for the values of the principal term, shewing that the values employed 

 by Hansen required correction, and that he had omitted a large 

 number of these terms, w^hich must be included before the lunar 

 tables could be brought into accord with observation. 



Third: The whole available series of observations of the moon 

 since 1620 were reduced to a uniform basis and compared with 

 Hansen's lunar tables, and their errors deduced for different epochs. 



Fourth : The long series of discordances between Hansen's 

 tables and observation were utilised for determining the corrections 

 to the elements adopted in Hansen's tables, and for deducing the 

 apparent value of the co-efficient, epoch, and period of every inequality 

 of long period shewn by the observations to exist in the complete 

 expression for the motion of the moon. 



Fifth: It was shown that when these corrections and additions 

 were made to Hansen's tables they sufficed to represent completely 

 the whole of the observed positions of the m(X)n since 1650. 



The whole of this work was completed and ready for publication 

 before the end of 1894. But the publication had to be deferred 

 because no funds could be obtained. Had it not been for this. South 

 Africa would have reaped the entire credit for the recent improve- 

 ments in our knowledge of the motion of the moon and have furnished 

 astronomy with the new lunar tables that are so much wanted. 

 Unfortunately, funds could not be obtained ; the publication had 

 to be deferred ; other astronomers, tired of waiting for these results, 

 were led to take up different portions of the work, and the credit 

 for priority in the recent advances in our knowledge of this important 

 branch of astronomy has passed away from South Africa. 



In 1898 the work of verifying and extending Hansen's theory 

 was taken up by Professor Brown in America, and brought to a 

 conclusion in 1905. The results obtained are in complete accord 

 with those obtained in Natal, though they are more extensive and 

 elaborate. 



In 1894 the extension of Hansen's calculation of the perturba- 

 tions due to the planets, was taken up by M. Radau in France, and 

 many of the new terms which had been calculated in Natal were 

 independently discovered and calculated. The results, so far as 

 thev go, are in accord with those obtained in South Africa, but 

 the calculation has not been pushed far enough to obtain the complete 

 values, so M. Radau's results are not in full accord with the results 

 yielded by the observations. 



In 1903 Mr. Cowell, of the Royal Observatory, undertook the 

 work of comparing the Greenwich observations of the moon made 

 between 1750 and 1902 with Hansen's tables, and employed the 

 observed discordances for determining the apparent errors of Hansen's 

 tabular co-efficients, as well as the apparent co-efficients indicated 

 by the observations for certain terms arising from the disturbing 

 action of the planets which had been calculated from theory by 

 M. Radau. The results obtained by Mr. Cowell are in general 

 harmony with those already obtained in Natal. It was found that 



