286 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



'=i-/:(4-.-^r<^- (x^-) 



provided that^ within the extent of these integrations, the radical does not vanish 

 nor become infinite. When this condition is not satisfied, we may still express the 

 simplified characteristic function w, and the time t, by the following analogous inte- 

 grals : 



w 



=/"±\/"^-^'^^. (Y^-) 



and 



in which we have put for abridgement 



(7 + T 



<r — T 



(119.) 



and in which it is easy to determine the signs of the radicals. But to treat fully of 

 these various transformations would carry us too far at present, for it is time to 

 consider the properties of systems with more points than two. 



On Systems of three Points, in general ; and on their Characteristic Functions. 



17. For any system of three points, the known differential equations of motion 

 of the 2nd order are included in the following formula : 



mi (.r"i S ^1 + 3/"i § t/i + s"i Iz^) +m2 (.^"2 ^ ^2 + 3/" 2 ^ I/2 + ^"2 ^ ^2) 1 



(120.) 



the known force-function U having the form 



V = m,m,/'''^ + m,m,/'''^ + m,m,/'''\ .... (121.) 



in which/^'' ^ , /^'' ^ , /^^' , are functions respectively of the three following mutual 

 distances of the points of the system : 



r^'' '^ = n/(^i - ^2)' + (yi - 1/2)' + (^1 - ^2)^1 



r^'' '^ = n/(^i - ^3)' + (yi - ^3)' 4- (^1 - %)^ > . . . (122.) 



r^'' '^ = n/(^2 - ^3)' + (y2 - ^3)' + (^2 - %)':_ 

 the known differential equations of motion are therefore, separately, for the point m^. 



o/'i = m2 -^^~~ 



,2) 



+ W.1 



8/ 



{h 3) 



S.rj 



y'i = 



'(1. 2) 



= ^ ?/l_^^ ?/ 



(1, s) 



m, 



^^1 



+ ^3 



8/(1. 2) 



mo — TT — + m^ 



gy.(l, 3) 



2 SSj ' ""3 8;2j 



(123.) 



with six other analogous equations for the points ^3 and m^ ; a/\, &c., denoting the 



