ORBITS RESULTING FROM ASSUMED LAWS OF MOTION. 



By Arthur Searle. 



Received Jan. 12, 1920. Presented Feb. 11. 1920. 



The present inAestigation attempts to solve some problems Avhich 

 were interesting to me, and which did not seem to be satisfactorily 

 treated in such works as I could conveniently consult. It is possible 

 that additional research among mathematical treatises would have 

 proved more successful; but in any case, it is not probable that the 

 precise method which I have here employed would have been adopted 

 by previous writers. The plan of inquiry, therefore, may be new, 

 even if its conclusions only confirm older discoveries. 



Under the law^s of inertia and gravitation, a particle of negligible 

 mass circulates in one of the conic sections about a central point from 

 which emanates an attraction proportional to the inverse square of 

 its distance. Denoting this variable distance by R and its inverse 

 square by i?~-, the question may be asked what effects will follow if 

 we substitute for — 2 an exponent n to which we may assign any 

 value, ^"arious cases of this kind are considered in ordinary text- 

 books, especially that in which n = + 1, but some which seem 

 particularly interesting are evaded. In attempting the solution of 

 these, I have been led to the general principles to be stated below. 



I. 



Transverse motion and the force derived from it. 



When the moving particle proceeds directly toward or directly from 

 the central fixed point, its course is controlled only by the velocity 

 attributed to it and by a single force. In other cases, its motion may 

 be resolved into radial and transverse components perpendicular to 

 each other. Its transverse motion produces a second force, called 

 centrifugal by established usage only when the orbit described is a 

 circle. If not, a distinctive name for this second force is required, 

 and if such a name is to be derived from an ancient language, the 

 words apocentric or divellent might be suggested. But, in the present 



