PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 283 



r2 = {rcos(^-cj-) — rocos(^o — '^)}^+ {r sin (^ — ter)— roSin(^o -cr)}2 

 = a2 {(cos V — cos VqY + (1 — ^^) (sin v — sin Vq^} 



= 4 a2siny^2 J / gin— ^"j + (1 — ^ ) vcos— ^) | 



= 4 a^ (1 — e/) sin y^^ : 



we may also consider r as having the same sign with sin v,, if we consider it as 

 alternately positive and negative, in the successive elliptic periods or revolutions, 

 beginning with the initial position. 



Besides, if we denote by o- the sum of the two elliptic radii vectores, final and 

 initial, so that 



<r = r + ro, . . (113.) 



we shall have, with our present abridgements, 



0- = 2 a (1 - e^ cos y^) ; (114.) 



the variables v^ e, are therefore functions of <r, r, a, and consequently the character- 

 istic function V, is itself a function of those three quantities. We may therefore put 



^ ' m^ + m^' V" •; 



w being a function of c, r, a, of which the form is to be determined by eliminating 

 V, e, between the three equations, 



w = 2 y^ (u, + e^ sin v),'] 



<r = 2a(l - e,cosy,), ^ (R) 



r = 2a(l — e^2)isiny^; J 



and we may consider this new function w as itself a characteristic function of elliptic 

 motion ; the law of its variation being expressed as follows, in the notation of the 

 present essay : 



In this expression, ^ yj ^ are the relative coordinates of the point m^, at the time t, 

 referred to the other attracting point Wg ^^ ^^ origin, and to any three rectangular 

 axes ; |' ?j' ^ are their rates of increase, or the three rectangular components of final 

 relative velocity ; a (3 y a' |3' 7' are the initial values, or values at the time zero, of 

 these relative coordinates lind components of relative velocity ; a is a quantity inde- 

 pendent of the time, namely, the mean distance of the two points m^ wig 5 ^^^ (^ is 

 the sum of their masses. And all the properties of the undisturbed elliptic motion 

 of a planet or comet about the sun maybe deduced in a new way, from the simplified 

 characteristic function w, by comparing its variation (K^.) with the following other 



form, 



V Stij. Sic. 8TOv^ .y„. 



