PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



119 



Mathematical Example, suggested hy the motion of Projectiles. 



24. If the three marks of position ri^ 712 % ^^ the moving point are the rectangular 

 coordinates themselves, and if the function U has the form 



g, (Jj, V, being constants ; then the expression 



n = iir,,^ + ^2'-\-^3')+g'i3 + i{(^''(^i' + ^2')+^^^3'h • . (111.) 



is that which must be substituted in the general forms (83.), in order to form the 6 

 differential equations of motion of the first order, namely, 



dflx drjQ drjQ 





o d tff^ o I 



(112.) 



dt — (" ''i' dt 

 These differential equations have for their rigorous integrals the six following, 



and 



7) 



?jj = e^ cos /[A ^ + — sin |M; ^, 

 n^ = €2 cos (Ji>t -^ — - sin (ju t, 

 7]^ = e^ cos p t -\--^ sinvt — -^ vers v t, j 



(113.) 



(114.) 



^1 ^^ Pi ^^^ ^t — i^j Ci sin |M< f , 1 



73^2 ^ P2 cos (Jb t — (j^ 62 sin (Ji^t, i 



tJ73 = /?3 cos V ^ — (i' eg + -7) sin f # ; I 



ej 62 63 jOj /?2 Pa heing still the initial values of jjj ri2 Vs ^^i ^2 ^3* 



Employing these rigorous integral equations to calculate the function S, that is, by 

 (85.) and (110.) (111.), the definite integral 



we find 



■BTc, + ■BTc 



+ U)6f/, 



(115.) 



')=i{/^l' + i'2'+i>/ + ^'(^l' + ^2') + G^3 + f)'}| 



- ^2 (ei2 + ^2')} cos2^^ - i^M, (^i;?i + eg/^a) sin2p^ >(116.; 



+ i{;^3' - (veg^- f y}cos2.^ - ^(ve, + f )j?3sin2./, 



and 



U = ^ - i { Pi' + ^^2^ + i^3' + i^' (^1' + ^2') + G e'3 + f )' } 



+ i {Pi' +P2' - i^^ (^1^ + ^2^)} cos2 (U, ^ - iit* (eijoi + e2;?2) sin 2^/ 5> (li;.) 

 ■^\\vi- Oe3 + ^yjcos2vj?-i(ve3 + f)/?3sm2.^; 



and therefore, 



