252 CELESTIAL MECHANICS. I. vii. 26. 



+Aq COS ^)+ Cj9d^ siD 6 cos ^ ; tbe sum of which is sin ^. 

 d (Br cos ^ + ^g sin <p) — cos ^.d (Br sin <p — Aq cos ^)4- 



dd.(sin25 + cos «^). sin ^ Cp-^di^ } cos d sin (p (Br sin ^ — 

 Aq cos ^) + cos fl cos <p {Br cos ^4->45r sin (p) ? + Cpd ^ 



sin d cos (pzz — Brd <p + d {Aq) -f d5 . sin p Cp -{■ di^ (cos 

 Br + Cj5 sin 9 cos «p) =: dq* + Cp (d5. sin (p + d>^. lin d cos 



^)— ^r(d?»— d^.. cos 6)izdq' + {Cpr'-Brp) df=] dg'+-^^ 



r'j3'd^=z~(dN'sin d-\-dN' cos d) sin ^+diV"'' cos (p. Lastly 

 [( — dN. sin & cos ^zrd9. sin cos 9 cos ^ (Br cos (p-\-Aq sin ^) 

 + sin 2^ cos 9.d (^r cos ^ + ^gr sin tp) — sin 9 cos (p d {Cp cos 

 6) ; — d N\ cos cos (p=:i— d5. sin cos 9 cos ^ {Br cos (p+Aq 

 sin ^) + d^// cos 9 cos ^ (^r sin^ — jiq cos ^) + cos2 5 cos ^d 

 {Br cos (p + Aq sin (p + cos cos (p.d {Cp sin 5); and — dN" 

 sin ^=— d%|/ cos 9 sin ip (Br cos ^+Aq sin ^) + sin ?>d {Br 

 sin 9 — ^gr cos ^) — Cpd ^|/ sin 6 sin ^ ; and adding- these 

 together we have cos ^.d {Br cos (p-^-Aq sin (p) — cos (p.d9. 



Cp + d-\^ cos 9 } cos <p {Br sin ^ — Jg cos (p) — sin (p {Br 



cos <p-\-Aq sin 9) > +sin ^d (Br sin (p — Aq cos <p) — Cp.d-^ 



sin 9 sin (p—d{Br)+AqA(p-\-co?> (p.d9. Cp—d-^» cos 5 ^g 

 + Cp d^ sin 5 sin ^ — dr' + Aq (d^ — d^. cos 9)—Cp 

 {d-\^ sin 9 sin ^ — d9 cos 9) = dr' + Aqpdt — Cpdt = ] 



d/+ '^tl^ p'q'dtzz—{dN sin fi + diV' C03 6) cos (p—dN" 



ALy 

 sin ^. 



Scholium. These three equations are very convenient 

 for the computation of the rotatory motion of a body, when 

 it turns very nearly round one of its principal axes, which 

 is the case with the rotations of the heavenly bodies. 



