698 CAVALLIN, SECÜLAR PERTURBATIONS. 



always holds good when 



\a,\>\a\>\a^\, (47') 



I a I may by the inodulus of a root or not, provided that | a^ | 

 and I «2 1 3'1'e situated sufficiently near to \a\. 



For, if I a j is not the modulus of a root, then as a con- 

 sequence of (43) 



M{a^) = M{a^) = M{a) . 



Particularly it follows from (47) and (47') 



2 



(48) 



with the condition 



I a, I > 1 > I «0 1 , (48') 



when I «j j and | a, | ^^'^ situated sufficiently near to unity. 



All the propositions we have deduced concerning Af(a) also 

 hold for G{a), as iraraediately follows by passing to the limit. 



Thus assuming the condition (48') it follows from (45) 



Ö(ai) ^G> G{a^) (49) 



and from (48) 



Assume in (1) that the right member consists of two terms, 

 and also that the condition 



\A,\ = \A,\ 



holds. 



Because g^ ~> y^i we have then 



and 



\A<'\<\A<\ 



and therefore from the lirst and second formula in (32) 

 (j(a,) — //, and G{a,^ -■= g^ , 



