204 SEARLE. 



When w = 2, the three terms found aboAe, the sum of which is 

 when the maximum is reached, become 



(A'—eY^-Y^) (A^-eY^) =A'-{l+e)A^Y'~-eA^Y^-\-c{] + c)Y^ (1) 



~iA^-eY^){eY^-\-Y^)= -{1+ e)A^Y^ + e{l-i-e)Y* (2) 



(A^-eY^- Y^Y-' = + eA^Y^-e{\ + e) Y^ (3) 



since then L = A, the semi-axis major, and Z = Y. The value of e 

 in this case was found above to be {A"^ — B'^)/B'^, in which B denotes 

 the semi-axis minor. 

 The equation determining the value of Y when R = P is 



A'- 2 (1 + e)^2p_|_ e {1+e) Y'= 0. 



The corresponding results from the use of R^dv as an equivalent for 

 2dt will next be deduced from the equation 1""= ii"'+ eZ'^= /?"'-[- 

 eR'"cos'"v, whence R'' = L^ / (l -^ e cos'^v) . To find dR, we have 

 wR""-^ dR= - D"ew cos""-! r d cos v/il + e cos"'?))^, and E"'-i = Z"'-V 

 (1 +f cos"'?0^"'"^V"', so that 



dR = - d cos v[eL cos"'-i »(l+f cos""?)) ("'-i)/«'-2]. 

 Also l/i?2= (1 + e cos«'i»)V"'/i', and 1/dv = -sinv/dcosv. 



Hence dR/RMv=e sin v L-^cos'^-hil+e cos'"!)) ("'-i)/"' -2(1 -ft- cos"'t))V"', 

 in which we may omit the factor eL'^, and differentiate the remain- 

 ing expression with respect to cos v to obtain the condition for the 

 maximum. 



It will here be considered sufficient to examine the results thus 

 obtained for the two cases w = 1 and w = 2. 



When w = 1, dR/R~dv = e sin v/L, which obviously has a maximum 

 when V = 90°, confirming the result less directly found above. 



W'hen w = 2, dR/R'^dv = e sin v cos v/Lyy{\ + e cos^t'). Omitting 

 e/L, and squaring, we require a maximum for (1 — cos^z;) cos'i'/ 

 (1 + e cos^z;). The squared denominator is always positive, and we 

 need to consider only the numerator of the differentiated expression. 

 It will be the sum of three terms 



— 2 cos'^r (1 + ^ cos^?)) (1) 

 2 cos V {\ -\- e COS"-?)) (1 — cos-<;) (2) 



— 2e cos^i' (1 — cos-?') ' (3) 



