678 SCHULTZ-STEINHEIL, THE ARGUMENT X^.- 



By summation of (19) from m = 1 to m = m we get 



^™ - ^0 =[C - cf] - 4c«' - c<»] + 1 46« - c] + 



m 



- 2 cos ,«^ (. [c« - ci'L J - (1 + «tC - c«_ J - 



' L ™ ™ — 1 J I 



2 -^ ™ 2 ™ 2 ' 2 '^ ™ 2*2'" 

 ^-I-S^J'-f-^Sr •• -(20) 



m = 1 nj = 1 



Here we have taken into consideration the relation 



r» = |C (21) 



a Telation that we get, if we remember that 



»"^"5 



„ ?>e ^, ., dO. 



2 ÖE 



where the right member does not contain any constant indepen- 

 dent of m. 



If we regard (16) we find that 



nöz = K„^ + r-C + etc.; 

 if we had not introduced 1 instead of e we had got 

 ndz = K'in + erC + etc.; 



that is, in ndz terms containing £- and growing indefinitely with 

 the quadrat of the time would have ocurred, but in this case 

 K'm would have contained terms multiplied by m-, which would 

 have equalled the secular terms in £-6"; it was for avoiding 

 the occurrence of those terms that Z was introduced. As X 

 always lies between and 7t, X-C does not grow indefinitely 

 with the time and in this case Km does not contain terms mul- 

 tiplied by ni^. 



