PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



143 



all the 3 ?z — 3 elements jM/^ co^ X^, and of their initial values (juq^ ^q t \) i> involving also 

 in general the time explicitly, we shall have the following forms for the 6 n — 6 

 rigorous integrals of the 6 n-^ 6 equations (S^.) : 



^i''i= dj. (^'*'*'); ^t''o,t 



^i'^i = Y^,(r''^>'')'^ ^t«o.t = 



_8_ 



-§-;r-(''^«'''); 



0,1 



'^i^i — fx. ^''^ *' ^^ 5 wz,. Vo^j = - g— (r, », J') ; 



i 0,t 



(Z'.) 



and in like manner we can deduce forms for the same rigorous integrals, from any 

 one of the eight combinations (Y^.). The determination* of all the varying elements 

 would therefore be fully accomplished, if we could find the complete expression for 

 any one of these 8 combinations. 



40. A first approximate expression for any one of them can be found from the form 

 under which we have supposed Hg to be put, namely, as a function of the elements 

 and of the time, which may be thus denoted : 



Hg — H2 {t, ;Ci, Xi, ^1, Vj, Tj, <yi, . . . «„_!, X„_i, ^„_i, Vn-\, '^n-li '"n-l) 



(A3.) 



by changing in this function the varying elements to their initial values, and em- 

 ploying the following approximate integrals of the equations (S^.), 



(B3.) 



For if we denote, for example, the first of the 8 combinations (Y^.) by G, so that 



G = {r, ;., .}, (C3.) 



we shall have, as a first approximate value, 



and after thus expressing Gi as a function of the time, and of the initial elements, we 

 can eliminate the initial quantities of the forms Tq xq vq, and introduce in their stead 

 the final quantities [ju cj X, so as to obtain an expression for Gj of the kind supposed 

 in (Z2.), namely, a function of the time t, the varying elements (/juX, and their initial 

 values (JbQ uq Xq. An approximate expression thus found may be corrected by a pro- 

 cess of that kind, which has often been employed in this Essay for other similar pur- 

 poses. For the function G, or the combination (r, «, v), must satisfy rigorously, by 

 (Y2.) (A^.), the following partial differential equation : 



