206 SEARLE. 



also be derived from the employment of UdZ for the equivalent of 

 2dt. It is needless to give the details of this process. 



When R = P, X = P sin v and Y = P cos v. It has just appeared 

 that A^ = B^ tan";;. Hence, from the equation B^X'- + A^Y^ = A^B\ 

 after division by B"^, P^ {sinh + tan*?' cos^?') = A^. Since tan''i' cos^d = 

 sin*»/cos"i', sin^» + tan^i' cos^« = sin-t'(l + tan-») = sin'r/cos^y = tan^tJ, 

 and P^ tan^i' = A'^. Accordingly, while the value of v increases as the 

 ellipse elongates, as appeared above, the ratio A/P also increases. 



Since tan^ij = X^Y'- = A/B, when R = P, AP'/B = A\ and 

 P^ = AB. Also sin-p + cos\' : cos-y = {A -\- B) : B, and cos-y = 

 B/(A + B). Similarly, sin^?; = A/(A + B). The position and 

 length of the radius vector P, when w = 2, are thus determined. 



As stated in section VI, cases in which w < 1 are not minutely exam- 

 ined in the present discussion. Bvit it may be remarked that since, 

 when w = 1, the radius vector P is perpendicular to the axis of 

 periastron, we may expect P to make with that axis an angle v, greater 

 than 90°, when w < 1. If no apastron occurs, the curve ^411 not 

 reach this direction of P at all, and the outward force will always be 

 in excess; if R ever assumes the Aalue P, it will subsequently attain 

 the value L, for which value the angle ^o will exceed 180°. 



X. 



Relative times of revolution in closed orbits of the same system. 



Since the momentary tendency to transverse motion is expressed 

 by the square root of the product of the radius vector and twice the 

 outward force, as appeared above in section I, and since the outward 

 force is expressed by mP^+^/R^, as in section III, the momentary 

 transverse motion is expressed by \^{2mP^'^/R^). The corresponding 

 increase of area is the product of this quantity by half the radius vector, 

 hence ■\/{viF"^^/2). The ratio of momentary increments of area in 

 two orbits belonging to the same system is therefore the ratio of their 

 respective values of \/P^+^. Let these values be denoted by \/(Pi''^) 

 and \/{P2^'^^), and let the corresponding areas traversed by the radii 

 vectores from periastron to apastron be denoted by Oi and ao. The 

 corresponding times of these angular movements are directly pro- 

 portional to the areas and inversely proportional to the momentary 

 increments of area, so that their ratio is aiV(-P2""^)/a2(V'-f*i"^)- 



