ORBITS RESULTING FROM ASSUMED LAWS OF MOTION. 199 



At periastron, the value of R is Q and that of Z is K. At other 

 points, therefore, (/?"' - (?'") = r(/v'^' - Z^'), which will be one 

 equation of the curve described by the moving particle. When 

 apastron occurs, i? = i and Z = 0, so that e = (X"' - Q'")/K'". 



Unless e' is infinite, e tends to approach as tv increases, and the 

 orbit ultimately becomes a circle. This is also evident from the 

 consideration that, with w indefinitely large, any radial movement 

 corresponding to a given transverse movement will instantly be 

 checked by the unlimited increase of inward force if the radial move- 

 ment is outward, or of outward force if that movement is inward. 



A second form of the equation of the curve, when apastron occurs, 

 is obviously (i"'- R"") = eZ"-\ in which, when R = Q, Z = K, and 

 c = (L«'- Q"')//v"' as before. 



Equations representing the curve may also be derived from the 

 quantities A', S, and H, corresponding, when measures are made along 

 the axis of apastron, to Y, Z, and A' in the system above employed. 

 The values of A" and S, like those of Y and Z, vary uniformly in the 

 Intervals between successive applications of force, so that dR/dX = 

 dR/dS, and is proportional to dR/dZ, but not equal to it except in a 

 circular orbit. 



By the method already explained, we find the ratio of the corre- 

 sponding variations of force on R and on S to be (n + 3) dR/dS; the 

 ratio of forces wdR/dS; and the resulting equations {L^— R^) = e 

 ^fjw_ gw>^ and i?"'- Q'"= e S«'. The value of e now becomes (L""- 

 Q^)/H^. Denoting the former \alue of e by Ci, and the new value 

 by 62, we find ro= eiK'"/m\ 



We found above {L""- R"") = aZ'", and m'c now find (D"- R"^) = 

 C2 {H""- S'"), whence (K'^/H'") {11""- S'") = Z'^, and //"' Z"- + 

 j^w gw = fjw j^w -yYjjgjj w= 2,H = A, the semi-axis major; K = B, 

 the semi-axis minor; Z = 1', and S = X. We then have the familiar 

 equation A^ Y^-\- B^ X^= A~ B~. When tc = 1, and apastron occurs, 

 as is here assumed to be the fact, II = L -\- Q and K = Q -\- L, the 

 value of each being the major axis 2 A; S = at periastron, and 

 Z = at apastron; the equation becomes S -\- Z = 2 A. 



By the substitution of R cos x for }', we convert the equation 

 (L^- R") = cZ^ into V" = R""^ e (R cos v - L cos vo)"", but unless 

 ir = 1 or 2, this form is less convenient than those above stated. 



