xl THE THEORY OF THE LONG INEQUALITY 



therefore, combining the two expressions just found, 



\36m i ri i aC 21mm'ria'(C - hC)\ 



K = - \ « ■ ," + , • t » — \ 



s l w' sin 1 w~ sin 1 J 



x a{D i e sin (\ + sr) + D 2 e'sin (\ + sr')} 



= - 0"-770 sin (\ + -&) + o"-211 sin (\ + sr) 

 = + 0"-898 sin\ - 0"-004 cosX. 

 The corresponding terms of $£, are 



J t J t de 2m n-a [ J t J t de J 



and SnV / / --rr 1 - - - -r-r/T 3w "° / / "V^f > 



.-. Sg = - o"'056 sin (X + w) + 0"-015 sin (X + ■&') 

 which may be neglected. 



These are all the terms of 3? and $<£', which appear to be worth retaining. 



48. The effect of the square of the disturbing force upon n may be calculated from 

 the value of $£. Let CsinftX + C'coskX be any term of $£, then 



S'n — —-^- = &a> sin l"(C cos &X - C'sin &X), 



the coefficients 6', C being supposed to be expressed in seconds. 

 Hence 8n = - o"-0000080 sin X + o"-0002343 cos X 

 - O"-O0OO319 sin 2X - 0"'0000722 cos2X 

 - 0"-00002<)l cos3X, 

 $ri = + 0"-0000058 sin X - 0"-0001709 cos X 

 + 0" -0000232 sin 2X + 0"-0000S2C cos 2X 

 + 0"0000212 cos3X. 



49- By (Art. 40), 



« ro dR , dR 



be = 2 f na* — - ±f, nae — ; 

 da 4J de 



therefore, representing the variation due to the square of the disturbing force by b'e, we have 



J, da 2 J,\ de de)' 



neglecting terms above the first order except such as will be divided by co 3 . 



The portion of bR to be retained is 



dRl ri 5 \ , dR ^ /S 

 -T- (die + b,e) + - — (b.w + 



de d-& 



dR, dR. dR s dR, , , dR, dR, '■ 



de dar de dsr de »' de S 



~de~ * l6 -^ + dw^' Br + ^ + d? ^ lC + - C) + d ' ^ + '*•? 



