200 SEARLE. 



VI. 



Angles formed by the axes of ijeriastron and apastron. 



The substitution of R cos v for }' is advantageous, however, in 

 finding the value of vo, when R cos v is taken as the independent 

 variable, since dZ = d Y. Hence the corresponding changes in R and 

 in cos V are reciprocal. 



At periastron, dR = dZ = 0, and the ratio dR/dZ is indeterminate 

 for the moment. The subsequent values of dR and dZ result from 

 forces which have the ratio w, as was found above, in section IV. 

 This ratio is independent of the order to which the infinitesimals of 

 time belong, since the time allowed for the action of the forces is the 

 same for dR as for dZ. The variations produced by these forces, and 

 the consequent changes in R and Z themselves, retain this ratio. 



While R varies from Q to L, Z varies from K to 0, and {K — Z) 

 from to K. The change in R is w times the change in Z. Hence 

 the corresponding change in cos v is l/w as great as that in Y, which is 

 equal to that in Z. 



When w = 1, dR/dZ is constant, and the total variation of cos i', 

 from periastron to apastron, is 2, from + 1 to — 1. In other cases, 

 the total variation of cos v is 2/w. When lo = 2, 2/io = 1 ; hence 

 cos V varies from + 1 at periastron to at apastron, and Vo = 90°, as 

 we know independently. 



For values of w between 1 and 2, Vo is obtuse, and cos i"o= — 

 (2/m> — 1). For example, when the inward force is inversely pro- 

 portional to the first power of the distance, tv = \/2, cos vo= — 

 (\/2 — 1), and the approximate value of vo is 180°— 65°. 5 = 114°. 5. 

 When the inward force is constant, «<' = \/'S; cos «o= — (2/V3 — 1), 

 and the approximate value of vo is 180°— 81°. 1 = 98°. 9. 



When iv>2, vq becomes acute, and if w increases without limit, Vo 

 approaches 0. We have already seen that in this case the orbit is 

 circular. 



When w<l, vo will exceed 180°, approaching 360° as ic approaches 0. 

 These cases will not here be minutely examined. 



The value of A' results from that of cos Vo by means of the equation 

 K = Q — L cos Vq, in which cos I'o is negative if I'o is obtuse. The 

 value of II is L — Q cos vo. The curve can accordingly be drawn 

 when ^•alues of Q, L, and w are assumed. 



When w>\, II> K. For // - A' = (L - Q cos vo) - {Q-L cos ro) = 



