Hence 



of Dynamical Problems. 



447 



(11). 



With this, and with co = l and m — b z , for simplicity in the 



numerical work which follows, we have 



d 2 x n du A . f b*\ 



-r* — 2-r = A = .r( o ~ , . . 



o¥+ l di 



Y=- V 



and 



? 2 = 2E + 3^ 2 + 



_ <T 



2// 



r N-2y 



From equations (12) and 

 (13), G.W.Hill has, with four 

 different values of E, found x 

 and y explicitly in terms t, for 

 the particular solution in each 

 case which gives the simplest 

 orbit (relatively to the revolving 

 plane XOY) ; of which the 

 one which presents the greatest 

 deviation from the well-known 

 1 variational ' oval of the ele- 

 mentary lunar theory is a 

 symmetrical curve wdth two 

 outwardly projecting cusps cor- 

 responding to the moon in 

 quadratures. He supposed 

 this to be the most extreme 

 deviation from the variational 

 oval possible for an orbit sur- 

 rounding the earth. Poincare, 

 in his Methodes Nouvelles de 

 la Mdcanique Celeste, p. 109 

 (1892), admiring justly the 

 manner in which Hill has thus 

 ' si magistralement ' studied 

 the subject of finite closed lunar 

 orbits, points out that there 

 are solutions corresponding to 

 looped orbits, transcending 

 Hill's, wrongly supposed ex- 

 treme, cusped orbit. Mr. Hill 



(12) 



(13) 

 (14) 



(15) 



Fig. 3. 

 T 



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