PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



101 



Reciprocally, if the form of S be known, the forms of these equations (C.) can be 

 deduced from it, by elimination of the quantities e or ;? between the expressions of its 

 partial differential coefficients ; and thus we can return from the principal function S 

 to the functions F and U, and consequently to the expression H, and the equations 

 of motion (A.). 



Analogous remarks apply to the functions Q and V, which must satisfy the partial 

 differential equations, 



"~ yr + * V^^l' "^25 • • • ^3n' bVi' Scr^' • • • 81:^3^ - ^ \8< 85T2' • • • B^^J' 



SQ 8Q 



3n 





>(32.) 



and 



^ VH' ^' • * • S^' ''^' '''^'"' ''^"Z = W + ^ ('?P ^2» • • • ''Isn)^ I 



^ \8^^ 8^' • • • 8^^ ^1' ^2. • • • ^3 J = H + U (ei, ^2, . . . egj. 



. . (33.) 



General Method of improving an approximate Expression for the Principal Function 



in any Problem of Dynamics. 



« 7. If we separate the principal function S into any two parts, 



Si + 82 = 8, (34.) 



and substitute their sum for S in the first equation (C), the function F, from its 

 rational and integer and homogeneous form and dimension, may be expressed in this 

 new way, 



„/8Si 8S1 \ T./SS2 _8S2 \ 



, +^uJh + "- + ^u,3„;8,3„' 



^^'(^:)=F'(lf)-F'(|5.), (36.) 



because 



and 



and since, by (A.) and (B.), 



