PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 137 



in which, if e be any of the first five elements, or the distance r, 



{e,^} = --7Vlg^ + ^8y +?87;,{e,?} = ~ jy,{e,;c}=0, . (193.) 



and 



-1 



8 CO /^\ li!i <?? Sctt Sett 



T? °" W/ ' 8r~""^T?^ "87—^' 094.) 



the formula (192.) may therefore be thus written: 



CO 8 ctf I 



+ K''} +-8x^^^^>+8]:^^^^>- J 



We easily find, by this formula, that 



{;.,.,} = - 1; {X,«}=0; {fi,^} = 0; {r,^}=-^|^; . . . (196.) 

 and 



8 V 8ctt 8a) 

 {".«> = -8i'r?-"8l. = ^ (197.) 



The formula (195.) extends to the combination {r, u) also; but in calculating this 

 last combination we are to remember that r is given by (Q^.) as a function of », jm», r, 

 such that 



87 = -^r' — • • 098.) 



and thus we see, with the help of the combinations (196.) already determined, that 



8t 8ett 8 /»»• 8 /»»• 

 {r.-) = -r.-l^ = r.J,®rdr + ^^J^€l^dr, (199.) 



if we represent for abridgement by 0^ and CL^ the coefficients of ^ r under the integral 

 signs in (Q^.), namely, 



M + m x^ ■) -* 1 



^ / M dr C . ^,, ^,, M + m xn-4 ] 



^ X / M. + m dr C y ^y,„ p, k M + w x« •) 

 "' = F\/Tr-7dl3{V + 2M/(r) M-'FJ 



These coefficients are evidently connected by the relation 



8e 8/2 



^ + Tir = 0' (201.) 



which gives 



■hX®rdr-\-~:Xn,dr = i>, (202.) 



r, being any quantity which does not vary with the elements » and ^ ; we might 

 therefore at once conclude by (199.) that the combination {r, »} vanishes, if a diffi 



MDCCCXXXV. T 



