Presidential Address. 2t, 



the observations indicated values for these co-efficients in close agree- 

 ment with those derived from theory by Hansen for the effects of 

 the disturbing action of the sun. He confirmed the conclusion that 

 the tabular place of the moon could not be brought into accord with 

 observation except by the introduction of a new term of very long 

 period, together with a number of smaller terms of shorter period 

 arising from the disturbing action of the planets. He shewed that 

 the observed values of the perturbation due to the planets differed 

 considerably from those assigned by M. Radau as yielded by M. 

 Delaunay's method of calculation, and that the observations yielded 

 much smaller values for the perturbation due to the figure of the 

 earth, than those obtained by Professor Hill, also by means of 

 Delaunay's method. Hence Mr. Cowell's investigation, like the 

 earlier South African ones, yielded strong indications of the incom- 

 pleteness of the values derived from theory by the use of Delaunay's 

 method of calculation as hitherto employed. 



Hence the present state of the lunar theory appears to be that 

 it is impossible to represent the observed position of the moon unless 

 there be introduced into the tables one or more new large terms 

 of long period which cannot be deduced from the known disturbing 

 forces by the use of the final equations in the form obtained by 

 Hill and Radau from Delaunay's method of developing the theory 

 of the moon. Hence if these equations be considered as yielding 

 the complete value of the terms of very long period, then, as urged 

 by Professor Xewcomb, " the cause of the observed changes of 

 long period in the moon's mean motion remains the greatest enigma 

 in gravitational astronomy." 



Yet the explanation of this enigma has been yielded by South 

 Africa, and it is to be found in the proof that has been given of 

 the incompleteness of the results yielded by the final differential 

 equations which have been employed of late. When the complete 

 values are obtained of the terms of long period due to the indirect 

 action of the planets, they are found to be much larger than those 

 vielded by Delaunav's equations as hitherto employed, and furnish 

 values in harmony with those yielded by the observations and published 

 as one of the last communications on the lunar theory dated from 

 South Africa. 



