Presidential Address. 21 



stated that according to his calculation the value of its co-efficient 

 should be only a I" instead of 2ii", and its greatest value should 

 be reached in 1859, and not in 1828 as supposed by Hansen. 

 Hansen denied that Delaunay's calculations were complete, and as 

 long as Hansen's tables were found to agree with observation, 

 astronomers were content to accept his conclusions. But when serious 

 discordances arose between the tabular and observed places of the 

 moon, and Professor Newcomb shewed that the tables were as errone- 

 ous for observations before 1750, as they were previous for observa- 

 tions since 1870, it became certain that Hansen's value for this 

 indirect Venus term must be erroneous. Here, then, seemed to be the 

 error which was destroying the accuracy of the tables of the moon. 



But when Delaunay's values were substituted for Hansen's, the 

 amended tables proved worse than Hansen's, and would not agree 

 with the observations for any period whatever. Yet Delaunay's 

 calculations weie certainly accurate, for they were verified by several 

 astronomers and found free from error. 



There had arisen a complete dilemma. 



For years no solution could be obtained. Then in 1886 an 

 investigation made in South Africa afforded a possible solution, by 

 shewing that though Delaunay's results were accurate as far as they 

 went, yet they were not complete, as he had not fully taken into 

 account the effect of the disturbing force of the sun on the minute 

 perturbation due to the disturbing action of the planet. But the 

 labour of calculating this omitted portion of Delaunay's investiga- 

 tion was so enormous that no one would undertake the work, and 

 as it involved only a great mass of small terms, the astronomers who 

 had verified Delaunay's calculation were of opinion that the 

 enormous work, even when performed, would only yield small correc- 

 tions to the values found by Delaunay. Hence they tried to explain 

 the failure of Delaunay's values to represent the observations as 

 due to the existence of yet other undiscovered large terms of long 

 period arising from the perturbation of the planets. 



Yet this explanation failed ; repeated investigation failed to 

 discover any other large term of long period, and it was shown that 

 if Delaunay's method of calculating these terms sufficed to yield 

 nearly the full value of these terms, then there could not be any 

 other large terms of very long period. Here was the dilemma worse 

 than ever ! 



As the Lunar Theory could not be left in such a state, the work 

 necessary to solve this dilemma was undertaken at the Natal 

 Observatorv. 



First: Hansen's theoretical values for the perturbations of the 

 moon due to the direct disturbing action of the sun were verified, 

 and it was shewn that when more accurate values were employed 

 for the moon's eccentricity and inclination and the solar parallax, the 

 results were in complete accord with those yielded by the observations. 



Second : An improved method was devised for calculating^ the 

 perturbations due to the planet, and approximate values obtained 



