ORBITS RESULTING FROM ASSUMED LAWS OF MOTION. 203 



We have first to express dR^/dZ^ in terms of Z by means of the 

 equation R"= V"- eZ"", from which wR"-^ dR = - weZ""-^ dZ, 

 and - dRIdZ = eZ^-^lR'"-^= eZ^-y{V"- cZ"-) ("'-!)/«', whence 

 dR^ldZ''= e^Z"^ ("'-D /(L"'- eZ'^f (""-i) /"'. 

 Since U'= {V'-eZ^YI^ -{Z + L cos v^f, 



c/i?V C^'^-^' = t'2Z2(«'-i)/[(L«' - eZ"')V"' - (2+i cos ?Jo)'] (i"" - eZ"')2("'-'V"'- 



We may omit the factor e^, which will not affect the result. 



The denominator of the expression obtained for dR^/UHZ'^ is 

 U2ji2{vj-\)_ It is always positive, and U = only at periastron. We 

 have therefore only to consider the numerator of the required differen- 

 tial expression. It wall consist of three terms, the differential of dR"^ 

 multiplied by U'^dZ'-, the differential of V^ multiplied by dR- and dZ'^, 

 and the differential of rfZ' multiplied by dR^ and V^. The second and 

 third of these terms will have their original signs reversed. 

 The first term is 2(w - l)Z2('"-i)-i dZ[{V"- eZ«')V"'-(Z + L cos v^f] 

 {V'- eZ"')2(«'-i)/"'. The second is - Z2("'-i) dZ{{V"- ^Z"')2(«'-i)/«'] 

 [2/w) {L^- eZ"')V"'-^ weZ^-^ + 2 {Z+L cos ro)]. The third is Z^C^-i) 

 [{V^- pZ^)V"'- (Z + Z cos v^Y] [(2 {iv - \)/io) {L^- (>Z-)2(«'-i)/«'-i 

 weZ'^-^dZ]. 



We may remove the factor 2Z2("'-i) -i dZ, which indicates minima 

 at periastron and apastron. The remaining factors may be reduced 

 to the form 



{w-1) [{L^- eZ"')V"'- (Z + L cos voY] {L""- eZ"')2(«^-i)/«' (1) 



-Z [a"'-eZ»)2("'-i)/"'] [(i"'-eZ"')2/«'-i fZ"--! + {Z+L cos vq)] (2) 



(w;-l)Z [(Z"'-eZ"')V"'- {Z+L cos v^Y] [(L^-cZ^)2(«'-i)/"'-i eZ"--!] (3) 



the sum of these three terms being when the maximum is reached. 

 For any given value of w, the corresponding value of Z at the max-imum 

 may be found, indirectly if not directly. From this value of Z, the 

 value of P is found by means of the equation of the curve. 



When 10 = 1, the expression to be differentiated becomes l/U'^, and 

 the proposed operation is not directly practicable. But in this case 

 dR/dZ is constant, so that we may regard R as the independent 

 variable. Since U = B. sin v, dU<dR except when U = R, that is, 

 when R is perpendicular to the axis of periastron, and an extreme 

 value of dU will then occur. The semiparameter of the conic section 

 is otherwise known to be the radius vector P, at which the outward 

 and inward forces are equal. 



