LECT. VI.] THE VARIATION. 25 



where we have written 



/3 = e'-me; 



this constant ft is associated with (lm)0 wherever the latter occurs; 

 for brevity in writing, we shall omit it. 



We may then assume as a first approximation 



au = 1 + a., cos (2 2m) 0, 



nt + e=0 + b 2 sin (2 - 2m) 6 ; 

 whence 



2(6- 6'} = (-2 - 2m) e - 2mb, sin (2 - 2m) 0, 



cos 2 (6 - 6'} = mb, + cos (2 - 2m) 6 - mb, cos (4 - 4m) 0, 

 sin 2(0- 6') = sin (2 - 2m) 6 - mb, sin (4 - 4m) 0, 



n~ = l + (2-2m) b, cos (2 - 2m) 0. 

 Substitute in the right-hand member of the second equation : 



-L -= = - 3m 2 [sin (2 -2m)0 + (2- 3m) b, sin (4 - 4m) ff\. 



Therefore 



/H-\ 3 m- . ., 3 2 3 HI .,, 



log, TT } - - cos (2 2m) + - - m-b., cos (4 - 4m) 0, 



> e \ lr I 2 1 m 4 1 m 



which we may write 



TJ2\ 



-TT) = 2A a cos (2 - 2m) + 2h t cos (4 - 4m) 0, 



where h is an arbitrary constant of integration, h n _ is a known quantity. 

 and h 4 involves b,. If we take as a second approximation 



au = 1 + a., cos (2 2m) + a, cos (4 4m) 0, 

 nt + e = + b^sm(2- 2m) + b t sin (4 - 4m) 0, 



the above value of \og e (H*/h 2 ) will not require modification and will 

 supply equations of condition for determining the coefficients a,, b,, ct t , b t . 



Thus 



dt n no? h I 



n d0~Hu' '' h H 



A. II. 



