PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



129 



C = - 



ih-PiY + (\ - P^^ (h - Psf 



2 [I tan /x. t 



2 V tan V ^ 



- ^ {i^l (^1 - Pi) + P2 (^2 - P2) + Ps (h - Pi) } 



+ - {Pi (>^i - ^1) + P2 ih - P2)} tan -^ + - Pi (X3 - ^3) tan ^ 



> (168.) 



- I iPi' + P2' + ?3^) + 7 (i^i^ + ^^2^) tan ^' + -j ;>/ tan ^ 



+ ("2 "• ^ + T ^^tan »' ^ ) g- (X3 - ?3) + (t ■" "7 ^^^ ^) ^i^3 



+ (^ - -6 - fv cotai^ ^ ^'' 



which may be variously transformed, and gives by our general method the following 

 systems of rigorous integrals of the differential equations of varying elements, (150.), 

 (151.): 



= - — = — ^'^P' - ^ tan — 

 1 S^j jttsin/x^ H* 2 ' 



^2 — — g^2 ~" H^sinju,^ y. ^"^^ 2 ' 



>. . (169.) 



and 



^ _ L5 _ _ hzUb _ P3 tan - -^ -^ f-^ -"l 



^3— 8^3 ~ vsinv^ V '' 2 ^ v Vsinv/ v /' 



«i = 1^ = - (Xi - p^) {t + ^cotan^^) +;^i (- ^ + ^ tan ^), 



^2 = 1^= - (>^2-;^2) (^ + ^cotan^^) + ;)2 (- ^ + ^ tan ^^), 



^3 = |r= ~ (^3-?3) 0+ ycotani/j?) +i?3(--^ + -7ta^V) 



MI70.) 



+ ^(¥~"^ + "r^°*^^''V' 



that is. 



and 



Xi = jOj cos jU» ^ — Cj |W/ sin ^ #, 

 X2 = P2 cos (jbt — 62(^1^ sin jfA /, 

 X3 = ^3 cos f # -- ^3 1' sin f ^ + ^ (^^ ;;- sin v tj, \ 



»j = ej (cos |a< ^ + jw, # sin ja» ^) + jo^ (^— sin p # — # cos ^ ^j, 

 X2 = 62 (cos |U» # + /M. ^ sin p #) + ;>2 ("j;: sin ^ # — # cos ^ t), 



«3 = 63 (cos V t ■\- V t ^inv t) '\- pz \j ^va.v t — t cos V tj 



(versvt t . ^ I ^\ 



(171.) 



(172.) 



MDCCCXXXV. 



