282 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



and _ 



t = \/^ ,(v — vq — esinv -{- esinvQ); (107.) 



we find that this expression for the characteristic function of relative motion, 



+ (±-l-)dr 



V 7' a r^ 



deduced from (A^.) and (B^.), may be transformed as follows : 



V^ = Wj 7^2 v — (y — fo + e sin y — e sin v^ : . . . (F^.) 



in which the excentricity e, and the final and initial excentric anomalies v^ t»Q, are to 

 be considered as functions of the final and initial radii r, r^, and of the included 

 angle ^, determined by the equations (106.). The expression (F^.) may be thus 

 written : 



N^ = ^7n^m^s/^{y,^-e^^mv^, (G3.) 



if we put, for abridgement, 



y, = ^^-^, e^ = e cos ^^-^ ; (108.) 



for the complete determination of the characteristic function of the present relative 

 motion, it remains therefore to determine the two variables t»^ and e^, as functions of 

 r Vq ^, or of some other set of quantities which mark the shape and size of the plane 

 triangle bounded by the final and initial elliptic radii vectores and by the elliptic 

 chord. 



For this purpose it is convenient to introduce this elliptic chord itself, which we 

 shall call 4: ''? so that 



•7-2 = r2 H-r2o — 2rroCOS^; (109.) 



because this chord may be expressed as a function of the two variables v^, e^, (involving 

 also the mean distance a,) as follows. The value (106.) for the angle ^, that is, by 

 (95.), for — 0Q, gives 



^-2tan"'|y/i±£tan|-j = ^o- 2tan"'|Y/i±£tanf j = «r, . (110.) 



w being a new constant independent of the time, namely, one of the values of the 

 polar angle &, which correspond to the minimum of radius vector ; and therefore, 

 by (106.), 



r cos {0 — vr) = a (cos v — e), r sin (^ — tjt) = a ^ i — ^2 sin v, ] 

 rQeos{0Q — 7!t) — a{cosvQ—e), roSin (^o — ^) = a^ri:72sinyo;J * 

 expressions which give the following value for the square of the elliptic chord : 



