V1U THE THEORY OF THE LONG INEQUALITY 



where 



aM, - -_ {**<» + 5. -^- - W -^- - «■ -^}, 



^f. - - 5 W» + 6« M' . »„. «£ - J «£ j , 



8 I da fla 2 da 3 J 



4 l da da } 



** " F6 r^ " ^ " ^ ^ " 9a -5?" " " "dor}' 



8 I da da 2 da 3 J 



„ « 2 f „ n , ai d6 o (S) db 2 W\ 



aM s = - U(6/*) + 6,«>) - 8 + a — f- + « — ~ 1 ■ 

 4[ da da } 



«3f. - - {WWyCW 4. 19a **-- 7« 2 -^- - a3 -£-J , 



a ^=r 6 { 10a -i- +8a2 -d^ + a3 ^-}' 



aM 12 = - U6 2 (i) - a — M , 

 4 I da J 



where b k ^ is the coefficient of cos k (9 — ff) in the expansion of 



{l -2acos(0- 0') + a 8 }-t. 



There are also terms of the second order involving 2\. For these terms, 



J? 2 = m'N^ cos (2\ + 2-nr) 



+ mNj* cos (2\ + 2-sr') 



+ m'N 3 ee cos (2\ + vr + nr) 



+ m'Ni sin 2 iy cos (2\ + 2 II'), 



where 



aN x = - - l44i 4 ft) + 14a — ~ + a 2 -~=- ( > 



8 I da da 2 J 



ffl JV 2 = - - {S8fc<*> + 14a -f— + a 3 — V [ » 

 8 I da da j 



oAi = - {426 3 (i) + 14a -2— + a 8 -jV } , 

 4 1 da da 1 ) 



a~ 



aJV 4 - - - $,«>. 

 2 



