XXXVI THE THEORY OP THE LONG INEQUALITY 



Then as in the last article, 

 3n °- a f f d (. SR "> m 9m' 2 rcV , 4« 2 . i(PP 2 - P'P,) sin 3X + (PP., + PP.J) cos 3\ \ 

 tt JJ t de ~4«>*sinl" ( + m a) \- 7 (PP 2 + P'P*) sin\ + 7 (P'P 2 -PP 2 ') cos \] ; 



.-. $%= + 2"-279sin3\-0"-421 cos3X 



- 16"-115 sin X - l"-883 cos A, 



2wi 

 and since ££'— 7 a*8C nearly, 



8g = - l"-662 sin 3X + o"-307 cos3X 

 + Il"-756 sin X + l"-374 cos X. 



43. Again let 



where R 3 = m' (P :i cos 3X - P,l sin 3X), Art. (33), 



« „ dR 3 dR 3 v , dR t v dRj „ , 



then, as before, 



( r d (3U) 39m' 2 n 4 a 2 f% 4>m 



"X( V= TSrfET* + ^*> {<" , . + **> -2X + (PP3'-P'P 3 )cos2X} 



21wi' 2 7i 4 tt 2 4w 



+ 8ft)4 sin p > (1 + -T « 3 ) f («*i - W) sin 4X + (PP S '+ P'P,) cos 4X} ; 



^ = - l"-006 sin 2X - 0"-090 cos 2X 

 + 0"-266 sin 4X - 0"-057 cos 4X, 



and |f = r a 2 ^ nearly ; 



.-. 8£'- + 0"-734 sin 2X + 0"-066 cos 2X 

 - 0"-194 sin 4X + 0"'042 cos 4X. 

 These are all the terms divided by co 4 which are worth calculating. 



44. Now let 



jp d #i»„ . dR i % „ . dR i s i , d ^i s . 

 eic = — — 6 2 e + — — 6 2 sr + — r d 2 e + — - d 2 -ar 



de dw de d-& 



dR 2 , d# 2 s dR 2 , dff 2 



^dR 2 . dR 2 . dR dR 2 , , 



+ -r-o 3 e + — - d 3 OT +-— ^ 3 e + — - ^ro- 

 de a^r rf e dxr 



dR 3 , dff 3 . di? 3 R , di2 3 . 



+ -r- V + — d 2 W + — - c 2 e + -T-, 6,w 

 de erar de dw 



•■« (f d 4 R ~ 



Jf/ t de 



{ (d£ d]\ dP dP^\ . 



9m'W ) [de de + de ~~de~) *™ 



) f££ *Ei df_ d J!*\ ( 



( \de de de de J j 



2w 3 sin l" 



