278 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



h, H^, H^p by the three conditions (F.) (T^.) (U^.). Thus, if we observe that the 

 distances r, Vq, and the included angle S^, depend only on relative coordinates, which 

 may be thus denoted, 



a?j — ^2 = ?3 J/i ~" 3^2 = ^} ^1 "" ^2 == C) ") 

 a^ — a2 = cc,b^ — b2 = p,Ci — C2 = 'y,J 



we obtain by easy combinations the three following intermediate integrals for the 

 centre of gravity of the system : 



^u* = ^u-^',P^'ii*=^i,-b,P^'„^-^„-c,„ (83.) 



and the three following final integrals, 



«'// ^ = ^// - ^u> ^'u ^ = 3//; - K ^„ t-^u- <^m (84.) 



expressing the well-known law of the rectilinear and uniform motion of that centre. 

 We obtain also the three following intermediate integrals for the relative motion of 

 one point of the system about the other : 



''=^!^ + ^8^' ^ • • • • (85.) 



and the three following final integrals, 



f__ ^^0 z.8^ -] 



(86.) 



/3' - ^Zo , i:? 



y = f gy - '^ g^ ; 



in which the auxiliary quantities h, H^, are to be determined by (P.) (T^.), and in 

 which the dependence of r, r^, ^, on f, f], J, a, (3, y, is expressed by the following 

 equations : 



rrocos^ = f a + ;j^ 4-(^y. J ^ '^ 



If then we put, for abridgement, 



A = — 4- ^ 15 _ ^ p _ — go , h 



r "^rnan^' "^ ~~ rrosind' ^ ~ rg +ronan^' • (^^-^ 



we shall have these three intermediate integrals, 



r = A|-Ba, ;j' = A;?-B|3, ^' = A^-By, (89.) 



and these three final integrals, 



a' = B|-Ca, /3' = B;?~Cf3, y' = BC-Cy, ...... .(90.) 



