DIFFERENTIAL EQUATIONS OF ASTRONOMICAL INTEREST. 
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the contrary, I believe that the solution of the differential equation above, arranged 
as a power series in a and b, converges for all finite values of these parameters, 
and that this is a consequence of a general theory of linear differential equations 
considered in papers* published by me in 1902. As this theory is capable of 
application to many other differential equations, as will be illustrated below by 
application to the equation considered by G. W. Hill for the motion of the moon’s 
perigee, I wish to deal with it here, repeating the argument in part. 
§ 12. Consider any system of linear differential equations, the n 2 coefficients 
= u A x x + ... + u in x n , (i = 1, 2 , ..., n ), 
u {j being functions of t. If these are considered only for real values of t, the 
properties which we require to assume are that, along a certain range which we 
shall suppose to include t — 0, these functions u t] are single-valued, limited, and 
capable of integration, the same being true of certain other functions derived from 
these by multiplications, and further, that certain infinite series, which we shall 
prove to be absolutely and uniformly convergent, are capable of differentiation, term 
by term. But in the majority of practical cases the coefficients u tj may be looked 
upon as the values, when t is real, of functions of a complex variable t. In this case 
we suppose a star region to be defined by lines passing to infinity from certain points 
in the finite part of the plane, which we call the singular points ; we suppose t — 0 
not to be a singular point, and the lines may be straight continuations of the radii 
joining the origin to these singular points. Within this star region, bounded by the 
lines in question, the functions u tj are supposed to be single-valued and capable of 
development by power series about every point, forming monogenic analytic functions 
in the usual sense. Taking then any region within this star region, we obtain 
solutions of the differential equations, with arbitrary values for t = 0, in the form of 
infinite series of functions, obtained by quadratures, which are proved to converge 
absolutely and uniformly within the region taken. 
The method of forming these solutions is extremely simple, involving only 
integrations and multiplications, but the way in which the work is arranged, though 
often of great utility, does not seem yet to find common acceptance, and some words 
must be given to it. 
* ‘Proc. Lond. Math. Soc.,’ XXXIV., 1902, p-. 355; XXXV., 1902, p. 339. See also the same ‘ Proc.,’ 
2nd Series, II., p. 293, where it is explained that the same idea had already been used by Peano 
and others. To me the method was independently suggested by the theory of continuous groups, 
‘Proc. Lond. Math. Soc.,’ XXXIV., 1902, p. 91. Poincare’s conclusions as to the convergence of 
astronomical series have been criticised by G. W. Hill, ‘ Coll. Works,’ IV., p. 94 ; but the point there at 
issue is different from that considered here. In connexion with an example considered by Poincare, 
loc. cit., p. 279, see Bruns, ‘ Astr. Nachr.,’No. 2606 (CIX., 1884), pp. 217, 218. Also Borei., ‘ Theorie 
des Fonctions ’ (1898), p. 27; Hardy, ‘Quart. Journ.,’ XXXVI., p. 93: ‘Proc. Lond. Math. Soc.,’III., 
p. 441, and the references there given. 
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