PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



293 



they may therefore be expressed as functions of nine independent quantities ; for ex- 

 ample, of four lines and five angles, r^'^ rj^'^ /^^ ro^'\ f^ 0q^'^ P ^o^'^,on which they 

 depend as follows : 



(1)2 



s^ = r,^'^ ' + r,^'^ ' - 2 r,^'^ r^^ (cos ^o^^^ cos ^o^'^ + sin ^o^^^ sin ^o^'^ cos /), 



(2) 2 , (2) 2 



(2) 2 , (1) 2 



(1)2 

 *5 = ^ J 



.(2)^(2) 



l(2) 



(2), 



2 r^'Vo'^' COS (^^'^ - V ). 



r^^^ (cos f^ cos P^ + sin P^ sin P cos /), 



> (131.) 



"1 — '^o J 



<i, = r'^' " + n'" ' - 2 r''> r„''> (cos /=' cos ^„('> + sin /"' sin <!„(" cos ,), 



d^ = rf^ = 4- ^C) ^ _ 2 r„<'' /■' (cos ^/'' cos «'" + sin ^„<'' sin /'> cos ,), 



rf, = r<« ^ + r/'l ^ - 2 r<" r„f" cos (o'" - ^„<"), 



the two line-symbols r^^^ r^^ denoting, for abridgement, the same two final radii vec- 

 tores which were before denoted by r^^' r^^' , and Tq^^ Vq^ representing the initial 

 values of these radii ; while P^ 6^^^ &q 6q are angles made by these four radii, with 

 the line of intersection of the two planes r^ r" \ r^ r ; and ; is the inclination of 

 those two planes to each other. We may therefore consider the characteristic function 

 V^ of relative motion, for any ternary system, as depending only on these latter lines 

 and angles, along wich the quantity H^. 



The reasoning which it has been thought useful to develope here, for any system of 

 three points, attracting or repelling one another according to any functions of their 

 distances, was alluded to, under a more general form, in the twelfth number of this 

 essay ; and shows, for example, that the characteristic function of relative motion in 

 a system of four such points, depends on the shape and size of a heptagon, and there- 

 fore only on the mutual distances of its seven corners, which are in number 



( — 2~ = ) 21, but are connected by six equations of condition, leaving only fifteen 



independent. It is easy to extend these remarks to any multiple system. 



General method of improving an approximate expression for the Characteristic Function 

 of motion of a System in any problem of Dynamics. 



19. The partial differential equation (F.), which the characteristic function V must 

 satisfy, in every dynamical question, may receive some useful general transfor- 

 mations, by the separation of this function V into any two parts 



MDCCCXXXIV. 2 Q 



