266 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



if then we establish, as we may, the three equations of condition, 



2 . m .r, = 0, 2 . my, = 0, ^ .mz, — 0, . (45.) 



which give by (40.), 



X.mx X.my S.mz 



""" = -S^' ^" = T^' ^" = ~^^' ......... (46.) 



so that Xii i/ii z,! are now the coordinates of the point which is called the centre of gra- 

 vity of the system, we may reduce the function T to the form 



T = T, + T„ (47.) 



in which 



T, = i2.m(^';+y;+0, (48.) 



and 



T„=4(^,;+y; + 02m (49.) 



By this known decomposition, the whole living force 2 T of the system is resolved 

 into the two parts 2 T^ and 2 T^^ of which the former, 2 T^, may be called the relative 

 living force, being that which results solely from the relative velocities of the points 

 of the system, in their motions about their common centre of gravity .r^^ y^, z,, ; while 

 the latter part, 2 T^^ results only from the absolute motion of that centre of gravity in 

 space, and is the same as if all the masses of the system were united in that common 

 centre. At the same time, the law of living force, T = U + H, (6.), resolves itself by 

 the law of motion of the centre of gravity into the two following separate equations, 



T^ = U + H,, (50.) 



and 



T,= H,; (51.) 



H, and H^^ being two new constants independent of the time t, and such that their 



sum 



H, + H, = H (52.) 



And we may in like manner decompose the action, or accumulated living force V, 



which is equal to the definite integraiyj* 2T dt, into the two following analogous parts, 



V = V, + V„, - . . . (E<.) 



determined by the two equations, 



V,=/„'2T,d< (Fi.) 



and 



y,=f,'2T,dL (Gg 



llielast equation gives by (51.), 



V.= 2H,^; (53.) 



a result which, by the law of motion of the centre of gravity, may be thus expressed, 



V. = s/i^n - «.)' + (3/. - hy' + (^// - ^//' • ^2 H, 2 m : . . . . (H^) 



