690 CAV ALLIN, SECULAR PERTURBATIONS. 



üsing the relation 



^— = 1 + 2z + 2z-' + ... + 2z»-^ + 2 —^ 

 i — z 1 — z 



we thus get 



t — I, 

 cot 



2m 



l t—l,, t — lj. i — l,. "I 



{ + i + Ii + (« — 1) i \ 



\e "" + e ™ + . . . + e "' j 



= ± « ± 2^6 



+ ni 



±2'^ r-~i;.< 



1 — e '"■ 



I i^ 

 where we take the upper or lower sign according as j e '"■ 



less or greater than unity. 



In both cases the reraainder vanishes tbr m = cxj and we 



obtain 



. a> _ t — lj. 



cot— r — - = + i + 2i2^e ™ . (20) 



2m ^ ^ ^ 



With the aid of the relation (20) the equation (16) is 

 transformed into 



'dt 



^M + ^E2'2 ^" ^ «"" ™ - - ^^" 2 ^~"' ^ «" ^ , (21) 



where 



iw ^ "hJii^iJjui: (21') 



2)n ' 



and Z' and /" each denote the number of roots the niodulus of 

 which is less or greater than unity, and the symbols of suni ma- 

 tion, 1' and ^", have reference to such roots respectively. 



From the significations of the integers V and /" it follows 

 that they must satisfy the unequalities 



0^1', l":^fx^=m^ — m„ . (22) 



