PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



26.5 





^217' + 



• + ^>^ XT"' 

 1 



n„, 





_/8T\ ifi 1. i^. 4. _, , ^^^ 



h 



and 



_8V 



;= - Ue'J +^U^i +^2 8^ + 





'3n + k' 



. (C.) 



rv 



+ A 



80, 



k 



+ A, 



S*, 



A 



8 0, 



(D'.) 



besides the old equation (E.). The analogous introduction of multipliers in the cano- 

 nical forms of Lagrange, for the differential equations of motion of the second order, 



by which a sum such as 2 . X g- is added to -y- in the second member of the formula 



(Y.), is also easily justified on the principles of the present essay. 



Separation of the relative motion of a system from the motion of its centre of gravity ; 

 characteristic function for such relative motion, and law of its variation. 



10. As an example of the foregoing transformations, and at the same time as an 

 important application, we shall now introduce relative coordinates, x^ y^ z^, referred to 

 an internal origin x,, y,, z,, ; that is, we shall put 



X. = x^. -f x^,, y. = y,. + y„ z. = z,. + z„ (40.) 



and in like manner 



a. = a,.-\- a 



//> 



k=h + b 



//' 



^i — ^/i "■ ^11 » 



together with the differentiated expressions 



x'. = x\. + x',, y\ = y\. H- ?/'„ 2'. = ;2'^. + ;2'„ 



. (41.) 

 . (42.) 



and 



< = <• + <. h\^h\^^h\, c>c',^+c', (43.) 



Introducing the expressions (42.) for the rectangular components of velocity, we find 

 that the value given by (4.) for the living force 2 T, decomposes itself into the three 

 following parts, 



2 T = 2 . m (^'^ + y'^ + ^") = 2 . m (a;'; + }ft + ^") 



+ 2 (x', ^.mx\^ y\, t.my\-\- z',, 2 . m z') + {x\f + y^! + ^ >!) 2 m ; (44.) 



