696 



CAVALLIN, SECULAR PERTURBATIO NS. 



can not be satisfied by any rot)t the modulus of which is 

 unity. 



From the unequalities (38) it t'ollows corresponding to the 

 roots (40) the unequalities 



QoA 



> 



Assuming 



a \' ' ' 



Qo, 



(41) 



these unequalities are transformed into 



Qo, 



(>0, 



^1 > 



Qo 



2-1 



Qa 



>1> 



^ I 



^'^X^l 



Qo, 



> 





Thus we se that the nuraber of roots in (40) possessing a 

 modulus greater than unity is 



1"{Qo^) = a, + o, + ... + G^_^ 

 and a modulus les than unity 



^' {^""i) ^ ^'i —{Qi + Q2 + --- + Qx) ' 

 from which thus follows 



= ^h — ^^[Qi + Q2 + --- + Q^_,) — Qoi (42) 



and 



= ^1 — 2(^1 + Q._ + ... + Q^ 



'Ul 



(42') 



If a has such a va lue that 



\Qo,\>\^\>\Qo,^,\ 



the unequalities (41) immediately show that 



l'(a) — l"(a) 

 = ^i, — 2(^, + Q, + ... +Q^). (43) 



A mere glance upon the relations (42), (42') and (43) at 

 once shows that 



