10] PERTURBATION OF THE ORBIT OF THE NOVEMBER METEORS. 197 



where 



D 2 = (A - a' cos uj + (B- b' sin u')" + C". 



If we ignore the square of e', we have 



and 



D- = A- + R- + C* - 2a' (A cos u' + B sin u'} + a'' 2 



= K- [ I 2a cos (u' /3) + or], 



if 



* 2 ( I + a 2 ) = A* + R- + C H + a'"; 



K 2 a cos /3 = Aa', K"O. sin /8 = Ba'. 



Hence 



.4 - (Ac' + a') cos u' + - a'e' ( 1 + cos 2u') , , 



dX = - - - , - ( *-, 



K 3 [ I - -2a cos (u' - /8) + a 2 ]- 



B Be' cos u b' sin u' + - b'e' sin 2u' , , 

 K 3 [ 1 - 2 a cos (M' - ft) + a-] 1 27r ' 



CCe'cosu' du' 



- - > "^ - 



/c :! [ 1 - 2a cos (tt' -&)+ a 2 ] 1 - 71 " 



and to obtain the complete values of X, Y, Z, these expressions must 

 be integrated between the limits u' = and u' = 2ir. Change the variable 

 from u' to (f> when 



then the limits of <j> are also and 2ir ; further write 



,=-&(, + &, cos <6 + b., cos 2<f>+ ..., 

 (l-2acos< + a 3 f 



so that 



, 1 T 2ir cosn(}>d<J> 



^ 

 then noticing that 



o-f 



Jo n-2 



sin n<f>d<f> 



