3 n^ 



PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 295 



T - T7 P-Il 111 1XL^ 1 



the function F being always rational and integer, and homogeneous of the second 

 dimension ; and being therefore such that (besides other properties) 



T-T4-T4.13^!X. ri\8V ST, 8V, 



T-Ti + T^ + ^gV^g^^+^gV^g— +... + -^g^^, (E^) 



S^Jl S>;i S'Jl S)33^ S),3^ S,,3^ 



and 



ST.SV, ST,SV ,_LTi.!V^_oT (C^\ 



gSV.g,, +g8V,S,, + "-+^_SV,8,3^-^^2. ...... .(C^.j 



By the principles of the eighth number, we have also, 



8T , ST , ^ T , ,__g . 



-gV -" ^1' 8V — ^2J- • • 8V — '^sn' • • • («-•; 



and since the meanings of y^, . . . ^'g „, give evidently the symbolical equation, 



we see that the equation (A^.) still holds with the present more general marks of 

 position of a moving system, and gives still the expression (B^.), supposing only, as 

 before, that the two parts of the whole characteristic function are chosen so as to 

 vanish with the time. 



It may not at first sight appear, that this rigorous transformation (B^.), of the partial 

 differential equation (F.), or of the analogous equation (T.) with coordinates not 

 rectangular, is likely to assist much in discovering the form of the part Vg of the 

 characteristic function V, (the other part Vj being supposed to have been previously 

 assumed ;) because it involves under the sign of integration, in the term Tg, the par- 

 tial differential coefficients of the sought part Vg. But if we observe that these un- 

 known coefficients enter only by their squares and products, we shall perceive that it 

 offers a general method of improving an approximation in any problem of dynamics. 

 For if the first part Vj be an approximate value of the whole sought function V, the 

 second part Vg will be small, and the term Tg will not only be also small, but will be 

 in general of a higher order of smallness ; we shall therefore in general improve an 

 approximate value Vj of the characteristic function V, by adding to it the definite 

 integral, 



2 Q 2 



