706 CAVALLIN, SECULAR PERTÜRBATIONS. 



di{t + 2ml7i) — ü^{t) = 71 {k{t + 2mlyt) — k{t)} , 



where l is sorae positive or negative integer or zero. 



In the same manner we get from (27), because the series 

 in the right raember after the term 3It has the period 27tm, 



d^{t + 2mlTc) — fi^{t) = 2mlTcM 

 and thus by comparison with the last relation 



k{t + %nl7t) — k{t) = 2mlM 

 or particularly 



k{2mhc) — k(P) = 2mlM. 



Here k{0) in so far is arbitrary, as it can signify a positive 

 or negative integer whichsoever or zero. Amongst these innum- 

 berable determinations we select 



^(0) = , 



from which by (64) follows 



" 1 



r = l 



and 



k(2ml7v) = 2mlM, {QQ) 



where k(2ml7t) has the same sign as IM. 

 Assuming t positive and 



2m(l + '1)5T > t ^ 2ml7t 



zmrt 

 and thus by {<6&) 



k(t) = 2mE^^---M. 



If t had been negative we should have obtained 



k{t) = —2mE^^-M . 



■ ^ 2vt7T 



