PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 281 



The characteristic function V^ of relative motion may now be expressed as follows : 



^=Xt7^ ■ I' («^-^ 



in which p is to be considered as a function of the extreme radii vectores r, Vq, and of 

 their included angle ^, involving also the quantity a, or the connected quantity H^, 

 and determined by the condition 



± dr 



r2 /^IJ- _ ±' 



V rp ap r^ 



that is, by the derivative of the formula (A^.), taken with respect to p: the upper 

 sign being taken in each expression when the distance r is increasing, and the lower 

 sign when that distance is diminishing, and the quantity p being treated as constant 

 in calculating the two definite integrals. It results from the foregoing remarks, that 

 this quantity p is constant also in the sense of being independent of the time, so as 

 not to vary in the course of the motion ; and that the condition (B^.), connecting this 

 constant with r Tq ^ a, is the equation of the plane relative orbit ; which is therefore 

 (as it has long been known to be) an ellipse, hyperbola, or parabola, according as the 

 constant a is positive, negative, or zero, the origin of r being always a focus of the 

 curve, and p being the semiparameter. It results also, that the time of motion may 

 be thus expressed : 



^— 8H^ — ^1^2 Sa' ^^ •>' 



and therefore thus : 



t^r . ^^" ; . (D3.) 



V r a r* 



which latter is a known expression. Confining ourselves at present to the case a > 0, 

 and introducing the known auxiliary quantities called excentricity and excentric 

 anomaly, namely, 



e=\/l~|-, (103.) 



and 



„ = eos-'(^-f^), (104.) 



which give 



4:^2a? — r^ — j9a = aesiny, (105.) 



V being considered as continually increasing with the time ; and therefore, as is well 



known, 



r = a (1 — e cos y), Tq = a (1 — e cos vq), 



a = -2tan-'{v/|^:ta„|-}-2ta„-'{v/ii-:tan.|}, | " 



106.) 



