20 Report S.A.A. Advancement of Science. 



The earlier tables of the moon could not be trusted to give 

 the true position of the moon to within 15" to 20" ; so that in 

 taking the moon as your clock you might be in error by thirty to 

 forty seconds of time, and so could not be sure of your position on 

 the earth by ten or more miles. 



But some sixty years ago new tables of the moon were calcu- 

 , lated by Professor Hansen at the cost of the English Government, 

 and they were published in 1857. These tables were found to be 

 far superior to any previous ones, for the places of the moon calcu- 

 lated from them were found to agree within three or four seconds of 

 arc with the true places, thus reducing the error of the moon when 

 used as a clock to some six or seven seconds of time, and the corres- 

 ponding uncertainty of the longitude of a position on the earth to 

 less than a couple of miles. This was rtiuch better than the previous 

 uncertainty of ten or twelve miles; hence these lunar tables of 

 Hansen's were generally adopted and have remained in use to the 

 present time. 



For ten years everything went on well, and the errors of the 

 tables remained small, but then they began to grow steadily larger, 

 till by 1875 the moon, regarded as a clock, was from ten to thirty 

 seconds fast, by 1885 the error had grown to twenty to forty seconds, 

 and by the end of the century was forty to seventy seconds too 

 fast. This was very serious, and it was obvious that there was 

 some error or errors in the mathematical basis of Hansen's tables. 

 But what were these sources of error ? 



The simplest plan for ascertaining this appeared to be to compare 

 Hansen's theoretical results with those which had been obtained by 

 Delaunay, and published some years after Hansen's tables. The 

 two results agreed. There were differences, but they were too small 

 to account for the large errors in Hansen's tabular place of the 

 moon revealed by the observations. Hence this comparison showed 

 that whatever might be the origin of these errors, it was not due 

 to mistakes in Hansen's calculation of the effects of the disturbing 

 action of the sun upon the moon. Critical examination of the basis 

 of Hansen's tables failed to reveal any possible explanation of the 

 origin of these discordances, except in the value that had been 

 deduced by Hansen for the effect of the disturbing action of the 

 planets. 



Hansen's tables contained two large inequalities of very long 

 period. One arising from the direct action of the planet Venus on 

 the moon with a co-efficient of over fifteen seconds of arc and a 

 period of 273 years, and the other due to the indirect disturbing 

 action of Venus with a co-efficient of over twenty-one seconds of 

 arc and a period of nearly 240 years. Hansen had found great 

 (lifficultv in calculating their values, and finally had adopted for 

 the second of these terms the value of the co-efficient and epoch 

 necessary to make his table agree with the observations between 1750 

 and 1850. Now, already in 1862, Delaunay had questioned the 

 accuracy of the values assigned by Hansen to this second term, and 



