112 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



as a function of the time and of the 6n quantities x^, ^g^ . . . x^^^, e^, eg, . . . 63^, and 

 observing that its variation, taken with respect to the latter quantities, may be shown 

 by a process similar to that of the fourth number of this Essay to be 



lE = y.(Xhx-p^e), (P.) 



we find that the rigorous integrals of the differential equations (G^) may be ex- 

 pressed in the following manner : 



in which there enters only one unknown function of elements E, to the search and 

 study of which single function the problem of perturbation is reduced by this new 

 method. 



We might also have put 



c==y^(-2.«^' + H2)rf<, .(7/.) 



and have considered this definite integral C as a function of the time and of the 6 n 

 quantities \ p^ ; and then we should have found the following other forms for the in- 

 tegrals of the differential equations of varying elements, 



^^- = + -fiX.' ^' = - 87. (L •) 



And each of t\\Q^Q functions of elements, C and E, must satisfy a certain partial differ- 

 ential equation, analogous to the first equation of each pair mentioned in the sixth 

 number of this Essay, and deduced on similar principles. 



18. Thus, it is evident, by the form of the function E, and by the equations (K^), 

 (G^), and (76.), that the partial differential coefficient of this function, taken with 

 respect to the time, is 



^ = ^!-2.|14^ = -H,. . . . .■■. . (M..) 



and therefore that if we separate this function E into any two parts 



El + Eg = E, . (N^) 



and if, for greater clearness, we put the expression Hg under the form 



H2 = H2 {t, «i, «2» • • • «3nJ ^1» >^25 • • • ^3n)j (^^ ') 



we shall have rigorously the partial differential equation 



8E1 8E2 / 8E1 , 8E2 8E1 , SE^x. ,_. • 



\ 1 1 3 n 3 n/ 



which gives, approximately, by (G'.) and (K^), when the part Eg is small, and when 

 we neglect the squares and products of its partial differential coefficients, 



^ |E, ^ ^X 



