PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



135 



if we put, for abridgement. 



8x 8a _ Sx 8X 8x Sa SxSa SxSx 8x8a 

 {«,A} — g^g^ g_^(g^ + 8>jSy "" gy 8,, + 8^8^ ~ 8^'8^» 



(Ps.) 



and form the other symbols {x,(ji^}, {\x}, &c., from this, by interchanging the letters. 

 It is evident that these symbols have the properties, 



{X,»} = - {«,X}, {»,«} = 0; (184.) 



and it results from the principles of the 15th number, that these combinations {«, X}, 

 &c., when expressed as functions of the elements, do not contain the time explicitly. 

 There are in general, by (184.), only 15 such distinct combinations for each of the 

 n— I binary systems ; but there would thus be, in all, 15 w — 15, if they admitted 

 of no further reductions : however, it results from the principles of the 16th number, 

 that 12 /i — 12 of these combinations may be made to vanish by a suitable choice of 

 the elements. The following is another way of effecting as great a simplification, at 

 least for that extensive class of cases in which the undisturbed distance between the 

 two points of each binary system (m, M) admits of a minimum value. 



Simplification of the Differential Expressions hy a suitable choice of the Elements. 



34. When the undisturbed distance r of m from M admits of such a minimum q, 

 corresponding to a time r, and satisfying at that time the conditions 



r' = 0, r">0, (185.) 



then the integrals of the group (P.), or the known rules of the undisturbed motion of 

 m about M, may be presented in the following manner : 



V = tan 



-1 »}2'-?y 



-X 





M + m* \/dr^ 



dr , 



rz=. dr 



> ■ (Q^O 



a; = V -|- sm 



^{2^ + 2M/(r)-(l + ^)^}' 



/W+m dr 

 V "~M 



-/QAx — A^ 



-X 



Vdr^'r 



-^ . dr 



y/{2..+ 2M/(r)-(l+^)^}' 



the minimum distance q being a function of the two elements «, f^, whicl must satisfy 

 the conditions 



2f. + 2M/(?)-(n-|)^ = 0,M/(y) + (n-H)J>0; ■ (186.) 

 and sin" ' s, tan" '<, being used (according to Sir John Herschel's notation) to ex- 



