ORBITS RESULTING FROM ASSUMED LAWS OF MOTION. 205 



The minimum occurs when cos v = 0, and in finding the maximum 

 the factor 2 cos v may be omitted. 



— cos'r(l + (' cos'-i") = — cos^r — e cos*iJ (1) 

 (1 — cos^i')(^l + (' cos'r) = 1 — cos-r + r cos~i — e cos^i" (2) 



— e eos-ij(l — cos-(") = — e cos"i'+ c cos^i' (3) 



and the required equation is 1 — 2 cos-?' — e cos% = 0, which may 

 also be expressed as sin^i' — cos\' — c cos^?' = 0, the value of e being 

 (A^ — B^)/B^ as before. It appears from this equation that as e 

 approaches 0, so that the orbit becomes nearly circular, the value of v, 

 when R = P, approaches 45°; and that it increases toward 90° as the 

 ellipse elongates. 



We may also refer the axial measurements to the major axis of the 

 ellipse by means of the equation R^ — B^ = e~X^, in which c denotes 

 the eccentricity of the ellipse, so that c^ = {A- — B^)/A^. Since A' = 

 R sin V, R' — 5^/(1 — e^ sin^?)). The value of R^dv is now R^d sin v/ 

 cos V = B-d sin v/ cos v (1 — c^ sin-i')- From the Aalue of R^ just found, 

 2RdR = 25V gin y d sin ^(1 - c'' sin^?;)^ = 2BdR/y/{\ - e" sin^r), and 

 dR = Be^ sin x d sin v/{\ — e^ sin hY/"^, whence 



dR/R\h = c^ sin v cos v/B\/{\ — e- sin\')- Omitting e^/B, a maxi- 

 mum is required for sin v cos r/\/(l ~ ('~ sin^v) or for its square 

 sin^»cos^t'/(l — e-sin^r). The numerator of the differential coefficient 

 is the sum of three terms. 



2 sin v{\ - sin^i') (1 - r^ sin^r) (1) 



- 2 sin»r(l - e" smh) (2) 



2 e" sin hi\ - sin^r) (3) 



The minimum occurs when sin v = 0. Omitting the factor 2 sin v 



(1 — sin-?') (1 — c- sw?t) = 1 — sin-r — c- sin^?- + e^ sin% (1) 



— sin-?" (1 — c~ sirvh) = — sin^r + c- sin^r (2) 



e^ sin-c (1 — sin-r) = + c- sin'-r — c~ sin^r (3) 



and the required equation is cos-t"— sin-r + '' sin^r = 0, in which 



We previously found sin^i- - cos^r - (.F - B^) cosh^/B'- = 0. The 

 sum of the two equations is {A- — B"-) sin"*('/.4- — {A- — B"^) cos"*^?'/ 

 B' = 0, whence sin^t' : cos'i' = A : B, and since sin-c = X'^jV- and 

 QO^H = Y^IF\ X"- : }'2 = A -.B when R = P. This equation may 



