199 



— = a x COS <p - a 2 SID 



Put for <p its first approximate value nt 



o» x cos <p = \ (A x cos (2n - n x t + e) + B x sin (2n - n x t + e) 



+ A x cos (n x t + e) - B x sin {nt + e)) 



<y 2 sin p = £ (-4 a sin (2w - n x t + e) - B 2 cos (2rc - ^ t + e) 



4- ^ 2 sin {n x t + e x ) + B 2 cos (^ + e)) 



taking the difference, and integrating, 



i = L + i [ ^ 1 + ^ 2 s i n (2n-rii t + e) + ~ — — cos (2n-n x t + e)^ 



+ J — sin (w^ + e) + cos (n x t + e) 



\ ^1 w l j 



for the variations of <p we have 



d<3 COS i , . x 



-~- = # 3 — - — sm <3 + « 2 cos <b\ 

 dt sinr r r/ 



its variations, therefore, will arise partly from those of &> 3) and partly 

 from those of a x and u % . We may set aside the former for the present, 

 and confine ourselves to those of a x and a 2 . 

 Then, priming the function 



cos t 



— (a x sm <p + eoo COS 0) 



sm i r r 



exactly in the same way as above, and integrating, we have 



, Icosi (A x +B 2 '- ^--Si . ^ x i) 



<p = nt + -- — {— cos(2»-w^ + e — sin (2ft -w, 1 + e)> 



2 sm t ( 2n - n x v 2n-n x v / ) 



1 cos* Ui-|5 2 , A 2 +B x . . . 

 + - - — < — — cos (n x t + e) — sm (n x t + e) 



2 sm t ( n x v y »f v y 



also since the differential equation for ^ is 



-TT = ~ — (^i Sin S + « 2 COS (3) 



the variations of will be the same as those of <p. only not multipled 

 by cos i. 



