220 Mr Brown, On the part of the 'parallactic [June 1, 



June 1, 1891. 

 Prof. G. H. Darwin, President, in the Chair. 

 The following communications were made to the Society : 



(1) On the part of the 'parallactic class of inequalities in the 

 moon's motion, which is a function of the ratio of the mean motions 

 of the sun and moon. By Ernest W. Brown, M.A., Fellow of 

 Christ's College. 



In a paper to be published shortly, a solution will be given by 

 approximation in series, of the equations for this class of in- 

 equalities. In Vol. I. of the American Journal of Mathematics, 

 Mr G. W. Hill has shown that by using rectangular instead of 

 polar co-ordinates, the inequalities depending only on the mean 

 motions of the sun and moon can be found to a high degree of 

 accuracy with comparatively little trouble ; and that the series to 

 be obtained may be rendered more convergent by developing in 

 terms of n'/(n — n) — m', instead of n'/n = m as has been done 

 in most of the previous theories. He further shows that by 

 developing in terms of m'/(l — ^m) = //. a still greater degree of 

 convergency is obtained. The results are expressed in rectangular 

 co-ordinates and on that account are not convenient for obtaining 

 the algebraical expressions of the longitude and parallax of the 

 moon. But these transformations will still have force when we 

 change to polar co-ordinates, so that by putting in Delaunay's 

 series for this class of inequalities m= m'/(l +m') = //-/(l +!/"■) 

 and expanding in powers of m or /u, we should get a better ap- 

 proximation to the truth. This is not necessary in the Variation 

 Inequality which Delaunay has found with sufficient accuracy for 

 practical purposes. But in the Parallactic Inequality he stops at 

 (a/a').m 7 , the numerical multiplier of which is roughly 55113; 

 the term expressed in seconds is 0"'38. By substituting 



m = m'/(l + mf) 



and developing in terms of m', this multiplier is reduced to about 

 one half its former value ; but since m' is nearly one-twelfth 

 greater than m the accuracy is not increased, though the new 

 series has greater convergency. 



On using Hill's method with rectangular co-ordinates for the 

 parallactic class of inequalities (i.e. those dependent on the ratio 

 of the mean distances of the moon and sun) a factor whose value 

 is 1/(1 — 4<m' — ...) appears, and to the expansion of this factor in 

 powers of m is due the slow convergence of the series giving the 



