PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



139 



its partial differential coefficient \f^J S^^r), obtained by treating q as constant, 

 vanishes at the limit r = 9- ; nor need it be varied with respect to q, because, by (186.), 



r. + srfi = ^-- • • ■ (209-) 



it may therefore be treated as constant, and we find at last 



{r,c,}=0, (210.) 



the two terms (199.) or (203.) both tending to infinity when r tends to q, but always 

 destroying each other. 



36. Collecting now our results, and presenting for greater clearness each combi- 

 nation under the two forms in which it occurs when the order of the elements is 

 changed, we have, for each binary system, the following thirty expressions : 

 {K,K} = 0, {k,(ju} = 0, {K,u} = 0, {»,r} = 0, {»,*;} = - 1, ^ 

 {X, «} = 0, {X, fA} = 0, {X, V} = 1, {X, r} = 0, {X, cj} =z 0, 

 (it*, k} = 0, {[Jt,, X} = 0, {fju, v} = 0, {^, r} = 1, {|M., 6f} = 0, 

 {v, k} = 0, {v, X} = - 1, {V, (I.} = 0, {u, r} = 0, {V, CO} = 0, 

 {t,k} = 0, {r,X} = 0, {T,(^} = - 1, {r,v} = 0, {r,<w} = 0, 

 {«, x} = 1, {ft;, X} = 0, {a;, /a} = 0, {*;, *'} = 0, {a;, r} = ; 

 so that the three combinations 



{/*,r} {&>,x} {X,v} 

 are each equal to positive unity ; the three inverse combinations 



{r,^} {»,&>} {v,X} 

 are each equal to negative unity ; and all the others vanish. The six differential 

 equations of the first order, for the 6 varying elements of any one binary system 

 {m, M), are therefore, by (O^.), 



(R2.) 



m 



m 



d fA 8 Hg 



dr 



8H, 1 

 8 Ho 



dt 



8(0 

 8H« 



> 





(S2.) 



£/^ ~ 8a ' 



and, if we still omit the variation of /, they may all be summed up in this form for 

 the variation of Hg, 



lU2=^.m([J^'^T-T'^(j^'}-aj'^»-»'^c^-{-X'^V'-v'hK), . . (T2.) 



which single formula enables us to derive all the 6 t? — 6 differential equations of the 

 first order, for all the varying elements of all the binary systems, from the variation 

 or from the partial differential coefficients of a single quantity Hg, expressed as a 

 function of those elements. 



t2 



