616 REPORT— 1895. 



this case on the ratio of the month to the year. The periods in the actual case, 

 however, must be corrected by terms dependino; on squares and higher powers of 

 the sun's eccentricity and the two parameters defininp: the amplitudes. 



In a paper in the ' Acta Mathematica,' vol. viii., Dr. Hill tinds the period of a 

 small oscillation of the first of the two kinds mentioned. His method involves 

 the consideration of an infinite determinant. He states that there can hardly be a 

 doubt that the determinant is convergent, but M. Poincare has submitted the 

 question to a rigid investigation.^ He concludes that an infinite determinant, 

 when the constituents of the leading diagonal are all unity, converges absolutely 

 and unifoi'mly if the sum of all the other elements is finite. Any determinant can 

 be reduced so that the elements of the leading diagonal are all unity, provided that 

 the product of these elements is finite. Dr. Hill's determinant satisfies these 

 conditions when the length of the month is sufficiently small. To complete the 

 proof it is necessary to notice that M. Poincar^, in his ' Mdcanique Ci5leste,' proves 

 that the series defining Dr. Hills variation curve converge for sufficiently small 

 values of the length of the month. 



At the conclusion of hispaper, Dr. Hill solved his infinite determinantal equation, 

 and obtained the principal part of the motion of the apse with great arithmetical 

 accuracy. The value he obtains difi'ers in the fourth significant figure from that 

 calculated from Delaunay's series ; it also agrees ■well with the observed value, 

 thus verifying a prediction of Delaunay's, as far as the apse is concerned, that the 

 remainder of his series would bring calculation into agreement with observation. 

 Dr. Hill has lately calculated an algebraic value to eleven terms for the principal 

 part of the motion of the perigee. He concludes by replacing the ratio of th© 

 month to the year by another parameter, empirically determined so as to increase 

 the convergence of the last terms calculated. This last step, however, does not 

 appear to be in any degree useful, as the convergency of the series near its tenth 

 term throws no light on the convergency of the remainder. 



The question of convergency of the series obtained in the lunar theory had 

 hardly been investigated before Poincare and Lindstedt. Formal solutions to the 

 seventh order and arithmetical solutions have been obtained, but it cannot be 

 assumed from the close agreement of the two that the coefficients can be repre- 

 sented by the algebraic series. Poincare has shown, however, that in certain cases 

 periodic solutions mu.st exist, and as a special case the series for the coeflicients of 

 the variational terms must converge for sufficiently small values of the ratio of the 

 month to the year. The motion of the node, so far as it depends on the ratio of the 

 mean motions only, had been investigated by Adams before Dr. Hill's work on 

 the perigee was published. Adams also obtained an infinite determinant. In the 

 arithmetical work, however, he used a dift'erent value of the ratio of the mean 

 motions to that used by Delaunay and by Dr. Hill. It is an illustration of the 

 almost unnecessary accuracy of the numerical work that it should have been 

 carried to fifteen decimal places, whereas the ratio of the mean motions, certainly 

 by far the easiest quantity to determine by observation, can only be depended upon 

 to seven places. I have recomputed the principal part of the motion of the node 

 using Dr. Hill's numbers. It may be noticed that the arithmetical value in this 

 case does not, as in the case of the perigee, justify Delaunay's prediction that the 

 remainder of his series would account for the discrepancy between theory and 

 observation. 



Dr. Hill's method of procedure is to use rectangular co-ordinates, the axes of 

 reference rotating round the ecliptic with a velocity equal to the sun's mean motion. 

 The calculation of the variation terms by this method is perhaps not so short as it 

 would be by some other method — possibly the best way to obtain them would be 

 by Delaunay's methods, the variables being reduced to two — but undoubtedly UO' 

 theory is so simple for the calculation of the higher inequalities. For each new set 

 of coefficients the problem can be quickly reduced to the solution of a system of 

 linear simultaneous equations. The principal parts of the motions of the perigee 

 and node are given by infinite determinants : the further approximations appear aa 



' Bulletin de la Societc Mathematiqve de France, xiv. pp. 77-90. 



