198 SEAKLE. 



V. 



Equations of curves derived from the preceding laws. 



Let us first consider cases in which w > 1. In these, an apastron 

 will ultimately occur. For, when R > P and is increasing, the 

 inward force is the stronger, and tends to retard the increase of R and 

 the decrease of Z. But if «' > 1, and the forces have the ratio ivdR/dZ, 

 the tendency to check the increase of R exceeds that checking the 

 decrease of Z. Hence dR will ultimately become less than — dZ, 

 however much it may have exceeded — dZ when R had the value P; 

 so that R will subsequently increase more slowly than Z decreases, Z 

 will reach the value before R reaches the value oo , and, by the defi- 

 nition of Z, R cannot further increase, and must attain its greatest 

 value L. The quantities A' and L must then be regarded as finite, 

 although we cannot set a definite limit to their increase. 



When IV < 1, the chance of an apastron diminishes. When w = 0, 

 it has already appeared that R always or never has the value P. 



We will now attempt to find functions of R and Z, the variations of 

 which, when corrected for the corresponding forces, shall have a con- 

 stant ratio. 



The ratio ivdR/dZ is a ratio of forces acting directly on dR and dZ, 

 and indirectly on R and Z themselves. The corresponding ratio of 

 forces acting directly on i?"'-i dR and Z'^-^dZ is i?"'-!/-^"'"^ times as 

 great, that is, wR^~^dR/ Z^'~^dZ. This is also the ratio of variations of 

 wR"" and of Z"', which is um'"-hlR/wZ''-hlZ, when we regard R and Z 

 as varying uniformly, undisturbed by any force. The ratio of forces 

 acting upon the variations is accordingly the same as the ratio of the 

 variations themselves when undisturbed by force. 



In all such cases, the existing ratio of the variations is not changed 

 by the application of the forces, not merely at a single instant, when 

 the forces are negligible as compared with the variations on which 

 they act, but after they have developed accelerating or retarding 

 effects upon the variations. Let A and B denote two velocities, a and 

 h forces acting upon them, li A/B = a/h = k, so that A = kB and 

 a = kb, (A + a)/{B + b) = k{B + b)/{B + b) = k. 



We conclude, therefore, that the variations of wR^ and of Z"' have a 

 constant ratio e' when the forces acting upon them have the ratio 

 wdR/dZ. Hence the variations of R"" and of Z"' have the constant 

 ratio e' /tv = e. 



