JS on-generating Force exerting Cumulative Influence. 1(35 

 2. The complete solution of the equation of motion 



a-\-/j?(l + 2 *Za r cosrnt)x = . . . (i.) 

 is given by 



00 



#= 2 A r sin{(c — rn)i+€} 3 .... (ii.) 



— 00 



where e is arbitrary and 



A r { / a 2 -( c - w ) 2 }+ y a 2 {a 1 (A r _ 1 + A, +1 )+« 2 (A,_, + A r+2 ) + ...}=-0. 

 On eliminating, the A's we obtain the infinite determinant 



. «, *i [-1] «!«,"';. |= ' • W 



where [r] denotes {/x 2 — (c — rn) 2 }/fju 2 . This equation deter- 

 mines c, and the roots are all included in the form +c — rn, 

 where r has the zero or any positive or negative integral 

 value *. 



Considering n as a variable parameter we observe that 

 the roots can become equal in pairs only through c becoming 

 equal to zero or half of some multiple of n. But the values 

 of n leading to equal roots separate the ranges of n giving 

 a real ' c ' from those giving a complex ' c/ Hence the 

 determination of the ranges of cumulative action is equiva- 

 lent to the problem of obtaining the values of n which 

 make the roots of the above determinant equal in pairs; 

 i. e. to the problem of finding the roots of each of the two 

 equations in n obtained from (1) by putting c = and } 2 n 

 respectively. 



3. When c = 0, [— f] = [ r ]» and when c=iw,[ — (r— 1)] = 

 [r] : on account of the symmetry thus introduced the idea 

 suggests itself that for these values of c equation (1) admits 

 of reduction. We shall investigate this question by direct 

 examination of the motion in the limiting cases. 



The general solution (ii.) may be written 



00 00 



.t- = cos e (sin ct% A r cos rnt — cos dS A r sin rnt) 



— 00 — 00 



-f sin e (cos ct 2, A r cos rnt + sin ct 2 A r sin rnt) . 



— 00 00 



Now in the limit when c = 0, either A_ r = A r or A_ r = — A r , 



* Thus far we follow the analysis introduced by G. Yv r . Hill in his 

 lunar theory. 



