268 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



we have, by difFerentiating these conditions, 



^.mx\=:0, ^.my\ = 0, 2.mz\ = 0, ..... (56.) 



...» * (57.) 

 (58.). 



(59.) 



11. As an example of the determination of these multipliers, we may suppose that 

 the part V,, of the whole action V, has been expressed, before differentiation, as a 

 function of H^ and of these other Qn—Q independent quantities 



•^/l ^,n = ?13 ^t2 '^/n = fe25 • • * "^/n-l ^^n = ^n-lJ 



Va - y.n = ^U 3/.2 - y,n = ^2, " ' V ,n-\ - Vm = ^«-l, > • • (60.) 



^/l — ^/« = ^IJ 2.^2 ~ «/n = ^2? • • • ^/n-1 " ^/n = 4-15. 



and 



(61.) 



that is, of the differences only of the centrobaric coordinates ; or, in other words, as a 

 function of the coordinates (initial and final) of n— I points of the system, referred to 

 the n^ point, as an internal or moveable origin : because the centrobaric coordinates 

 •^/,»3^/,'^/i'^/i'*/i' ^/i» ™^y themselves, by the equations of condition, be expressed as 

 functions of these, namely, 



^'i -Si - -Y^, i/,i - P7i ^r^, ;s,i - Ci - -Y^, . . . (62.) 



and in like manner, 



S.mu X .m^ X.my 



Xm 



^^^ = ft--T^' ^^ = y^--i^' • • • (63.) 



in which we are to observe, that the six quantities |„ 7\^ ^ «„ |3„ y„ must be considered 

 as separately vanishing. When V^ has been thus expressed as a function of the cen- 

 trobaric coordinates, involving their differences only, it will evidently satisfy the six 

 partial differential equations, 



.g... .g... .-i.., 



2'^;=.. .g.o, .'5=./ ' ■' 



