OF THE MOTIONS OF A SOLID BODY. 251 



dp' + --^ .^T'd^^dAT. COS d—d N\ sin ; 



dq' + -Yr^'rpdt—^(dN. sin 5+dJV'cos 9) sin^ + 



dN", cos (p ; 



A—C 



dr + pYd^zi— (diV. sin 0+ djV'. cosfi) cos^— 



dN''. sin ^. (J)) 



The equations for 37" afford us, by taking their fluxions, 

 when sin ^^=0, and cos tJ'^I, 



d5. cos 6 {Br cos <p + J^^ sin 9) + sin d.d {Br cos (p + ^gr 

 sin (p) — d {Cp cos 6) = — diV; 



d^)/. (jBr sin (p — Aq cos (f) — dd. sin fi {Br cos ^ + ^ry sin cp) 

 + cos 5 d {Br cos (p -^ Aq sin (p)4-d (Q) sin 0)=— diV^'; 

 d {Br. sin ^ — Aq cos (p) — dt^. cos 9 {Br cos (p + ^g' sin (p) 

 — Cpd4.sind =— diST^ 



Hence we have [d N. cos 0-n — d5. cos ^5 {Br cos (p-t-^g 

 sin (p) — sin cos Q . d {Br cos ^ + Aq sin ^) + cos 5. d ( Cp cos 5), 

 and— diV^'. sin 6= — d5 . sin ^5 (^r cos ^ + ^gr sin <p)+ sin 

 cos 5 . d {Br cos (p -\- Aq sin ^ + sin 5 . d (Q? sin d) + dx^. 

 sin fi ( Br sin <p — ^g cos ^), and adding these together, 

 •the sum will be — d5 {Br cos (p + Aq sin ^ + d {Cp) +d%^. 

 sin fi {Br sin ^— ^^ cos ^) + d {Cp)—Br (sin d sin ^. dil-— 

 cos^. dd) — ^^ (sin <p .d5 + sin 5cos ^. A-^)— Brqdit—Aqrdit 

 + d/, and d/+ (5 — -4) ^rdizz] d N, cos 5 - ^W. sin 



= dp'+ — qVdit, In the next place [ — diV^. sinfl 



sin ^=:dfi . sin cos 6 sin <p {Br cos ^ + ^5r sin ^) + sin ^ sin 

 ^ . d(Br cos^ + 7l^sin<p)— sinfi sin^. d (Cpcosfi) ;— d3^'. 

 co^^ sin (pzz — dd . sin cos 6 sin <p {Br cos <? + -4^ sin ^) + 

 d^^.cos 5 sin (p {Br sin ^ — Aq cos ^) + cos ^d sin ^ . d {Br 

 cos ^ + J.gsin^) + cos5 sin<p.d (Q)sinfi); + d W cos ^= 

 d>j^ . cos 9 COS ^ (Br oos ?>-|-^5' sin (p) — cos ^.d {Br sin p 



