ORBITS RESULTING FROM ASSUMED LAWS OF MOTION. 201 



(L — Q) (1 -\- COS Vo), and cos vo cannot be less than — 1. If w = 1, 

 H = K, the value of either being the major axis of the ellipse if 

 apastron occurs. If not, i = oo , and (L — Q) (1 -\- cos vo) becomes 

 indeterminate. In either case, e = (L — Q)/K = 2C/2A = C/A, 

 in which C and A have their usual meanings in the equations of a 

 conic section, and e denotes the eccentricity. 



When w = 2, cos Vo= 0, while H and K are respectively equal to 

 the major and minor semi-axes of the ellipse. If the curve is referred 

 to the axis of periastron, e = (A^— B^)/B^, in which A and B have 

 their usual meanings of the two semi-axes, and e may have any value. 

 If the curve is referred to the axis of apastron, e = {A"^— B'^)/A^, and 

 cannot exceed 1, being the squared eccentricity of the ellipse. 



VII. 



Relative distances of points on the curve from the axis of periastron. 



Since i?^= (L"'- fZ"') and Y = Z + L cos I'o, while R''= U^-\-Y^ 

 [72= (L"'- eZ"') 2/"-- (Z-}-i cos z>o)^ By differentiation with re- 

 spect to Z, d{U^) = {2/w) (L«'- eZ"") V""^ {-ew Z""-^) dZ - 2 

 (Z + L cos vo)dZ, whence d(U^)/2 = - [(L""- eZ^) V""^ eZ"'-i+ 

 {Z -\- L cos ?'o)] dZ. When R = Q, dZ — 0, and a minimum of U^ 

 and hence of U occurs, as is obviously the fact. 



When vq is obtuse, a maximum of U^ and hence of U occurs if 

 (Z + L cos Vo) = - iV"- eZ"^) V"-^ eZ'^-K Since (Z"'- cZ"') =R'^, 

 it is positive; Z and e are also positive. The maximum occurs, then, 

 when L cos Vo, which is negative, numerically exceeds Z. 



In the ellipse, when w = 1, i = .1 + C, e = C/A, and fo= 180°. 

 The condition for a maximum of [^ becomes Z — A — C = — [(A-{-C) 

 - CZ/A] C/A. Multiplying by A'' and reducing, (A'- C) Z = A- 

 (A -\- C) — AC {A + C), whence Z = A, and the maximum of U is 

 B, as is otherwise known. 



When w = 2, Vq = 90° and L cos ro = ; the condition for a maxi- 

 mum of U is Z = — cZ; hence Z = 0, and the maximum of U is A, 

 as is otherwise known. 



When w>2, {Z -{- L cos dq) is positive, and no maximum of U, in 

 the strict sense of that word, will occur. The greatest value of U will 

 be L sin Vq. 



