(V4.) 



294 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



Vi + V2 = V {V\) 



For if we establish, for abridgement, the two following equations of definition, 



analogous to the relation 



. T=..^((ii)v(^y+(m -(w^-) 



which served to transform the law of living force into the partial differential equation 

 (F.) ; we shall have, by (U^.), 



T - T 4- T 4- :S -{^-^' ^^ -i- ^^1 ^^^ 4- ^^^ ^^'A rY4 N 



A — li-h i2-t- ^-^^ VS;r 8^ "T" 8j/ ly + 1717/ » ^^•>' 



and this expression may be further transformed by the help of the formula (C), or 

 by the law of varying action. For that law gives the following symbolic equation, 



the symbols in both members being prefixed to any one function of the varying coor- 

 dinates of a system, not expressly involving the time ; it gives therefore by (U*.), (V*.), 



2.1ffi"i4."L."i + "^.^A=,^V,_ ..... .(Z4.) 



m \h X hx ' by by ' bz bz/ at ^ ^ ' 



In this manner we find the following general and rigorous transformation of the 

 equation (F.), 



^' = T-Ti + t,; (A5.) 



T being here retained for the sake of symmetry and conciseness, instead of the equal 

 expression U + H. And if we suppose, as we may, that the part Vj, like the whole 

 function V, is chosen so as to vanish with the time, then the other part Vg will also 

 have that property, and may be expressed by the definite integral, 



y2=f\T^T,Jr'l\)dt . . . (B5.|^ 



More generally, if we employ the principles of the seventh number, and introduce 

 any 3 n marks ?ji, ri^, . . . ^^^, of the varying positions of the n points of any system, 



(whether they be the rectangular coordinates themselves, or any functions of them,) 

 we shall have 



t = f(|^, l^,...p-\ (C5.) 



and may establish by analogy the two following equations of definition, 



