700 GAV ALLIN, SECÜLAR PERTURBATIONS. 



Also the M correspondins; t o the factor e "* is — ^ . 



m 



Then if /', l" and V" denote the number of roots, whose 



inodulse are respectively less, greater and equal to unity, we get 



in consequence of (51) 



M 



^n 7, 1 ,„ rv ,,„ 1 



= — + /'•—+ Z" • + /" • ^r- 



m ni zm 



_ 2m„ + 21' + l" 

 2m 



m^ + nin + l' — /" 



2m 



because 



/' + /" + /'" = mj — riin. 



We will noAv give general formulas for tlie calculation of 

 M{a). 



Changing the coefficients A into the coefficients (39') the 

 roots (11) of the equation (10) are changed into the roots (40) 

 of the equation (39). 



In stead of (21) we then obtain according to (18) and (19) 



^yae"^) 



Rcp\ 

 = M(a) + — Rl'^ >,a- "e ™ e '" R2'' Tj^^H ™ e "^ 



n = l 11=1 1 



where the symbols of sumraation, 2"'' and ^^'\ operate on those 



terms respectively in which | e ™ | is less or greater than unity. 



Tn this equation we put successively ao^, ao^, . . ., aOk in 

 steed of a, where Oy, (72, •-., Ok are the h roots of unity, and 

 add all the equations thus formed. 



Owing to the properties of symmetrical functions of roots 

 of unity, and because 



M{aOy) --= M{aß.,) = . . . = M{aOk) = M(a), 

 we thus obtain 



