100 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



SO that if we consider Q as a function of the 6 n quantities w. p. and of the time, we 

 shall have 6 n expressions 



. ^i = + 8^.' ^i = - 8^' (26.) 



which are other forms for the integrals of the equations of motion (A.), involving the 

 function Q instead of S. We might also employ the integral 



V=/'2.-|^<i< = 2/'t.rf,, . (27.) 



which was called the Characteristic Function in the former Essay, and of which, when 

 considered as a function of the 6 w + 1 quantities ri. e. H, the variation is 



lY,=:^{mlfi - pie) -\-tlli (28.) 



And all these functions S, Q, V, are connected in such a way, that the forms aii^d 

 properties of any one may be deduced from those of any other. 



Investigation of a Pair of Partial Differential Equations of the first Order, which the 



Principal Function must satisfy. 



6. In forming the variation (23.), or the partial differential coefficients (13.), of the 

 Principal Function S, the variation of the time was omitted ; but it is easy to calcu- 



late the coefficient -^j corresponding to this variation, since the evident equation 



dt — dt ^ -^ ^ri dt ' ' ' {^^') 



gives, by (20.), and by (A.), (B.), 



as ^, ^ 8H 



lT = S'-2...z.g^= -H (30.) 



It is evident also that this coefficient, or the quantity — H, is constant, so as not 

 to alter during the motion of the system ; because the differential equations of mo- 

 tion (A.) give 



dt — ^ \^ri dt "^ '^'ST dt) — ^ W^-) 



If, therefore, we attend to the equation (17.)> and observe that the function F is neces- 

 sarily rational and integer and homogeneous of the second dimension with respect to 

 the quantities w., we shall perceive that the principal function S must satisfy the two 

 following equations between its partial differential coefficients of the first order, 

 which offer the chief means of discovering its form : 



as . ^/dS SS 8S \ TT/ si 



^-F^^ ^ ^ e e e \--U(e e e ) ^^^'^ 



