PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



291 



evsvsvsvsv 



8V 



S«a 



' 8 a, ' 



8V8V8V8J^8V_ 

 oj/i + 8;^ "T- 8j/3 "t" 86i ■*" 8^»2 "*■ 8Z»3 - "^ ^ 



8V; SV 8V^ 



Szi + 8«2 "•■ 8^3 + 



8^1 



(P.) 



we find that these equations become identical, because 





0. 



(Q^.) 



But substituting, in like manner, the expressions (O.) in the equations of the form 

 (P.), of which the first is, for a ternary system. 



8V 



ev 



8V 



8V 



^ify^~^i^x, + '^2S3^^ —3/28^^ + -^38 



, 8V 7 SV , 8V 



8^1 



8«i 



'2 8 6. 



8V . 8V 

 -\-a 



2 8 a, 



(R4.) 



'3 8 6, 



and observing that we have 



X 



8V 



sv 



8V, 



8V, 



,, ~3/// §^„ -r «// gA ^'1 8a„ — '^^ 



(S^.) 



along with two other analogous conditions, we find that the part V^, or the charac- 

 teristic function of relative motion of the ternary system, must satisfy the three fol- 

 lowing conditions, involving its partial differential coefficients of the first order and 

 in the first degree. 



_ 8V, 8^, 



8V, 8V 8V, ^8V,, 8V, . SV, 



- ^2^ + aire - /^iS^ + «2gJ^ - /3 



'Ja 



^ — ^1 8 ?. ~ ^1 8 ri, "•" ^2 8 r ~- "02 s 



^2 

 8V 



8/3i 



Hi 



*Ii 





^18^1 ~'"^28y2 



28*2' 



8V, V 



^2 8^' ■ 



«-Ci 8^, ~^i8?i "T"^2 8^2~^^S?2 "^^I8«i~^i8yi +/2 8«2""'^28y^' 



(Tl) 



which show that this function can depend only on the shape and size of a pentagon, 

 not generally plane, formed by the point m^ considered as fixed, and by the initial 

 and final positions of the other two points m^ and 1712 ; for example, the pentagon, of 

 which the corners are, in order, m^ (m^) {m<^ m^ m^ ; {m^ and {m^) denoting the 

 initial positions of the points 7)i^ and m^y referred to 7n.^ as a fixed origin. The shape 

 and size of this pentagon may be determined by the ten mutual distances of its five 

 points, that is, by the five sides and five diagonals, which may be thus denoted : 



m- 



(mi) = V-^i, (mi) (mg) = Js^, (m.^) m^ = ^/*3, mgmi = 



= >sM,mi(mi)=yfi?5;/ 



^3 (^^2) = \/di, (mi) mg = ^d2, (mg) mi ^Jd^^.m^m^. 



the values of s^. . . d^ as functions of the twelve relative coordinates being 



