ORBITS RESULTING FROM ASSUMED LAWS OF MOTION. 193 



outward force. Any reader familiar with the elements of orbital 

 motion will readily fill up the outline of this demonstration by adding 

 details which it seems needless to state here. 



II. 



The relation of outward and inward force, token 7i> — 3. 



The present discussion will be confined to those cases in which the 

 exponent n, defining the law of inward force, as above stated, is at 

 least not less than — 3. I'nder this restriction, there may be, and 

 if ?i> — 3, there must be, with the exceptions noticed at the end of 

 section IX, some radius vector, to be denoted by P, at which the 

 outward and inward forces are equal. Under the law of gravitation, 

 this radius vector is the semiparameter of the orbit, as appears from 

 the ordinary treatment of the subject. For values of 7i other than 

 — 2, it is to be found by methods hereafter to be described. For the 

 present, it is required only to show that it will occur. 



Suppose that in a particular case, and at a given moment, the 

 inward force is the stronger, while at the same time the radius vector 

 is diminishing by the operation of the law of inertia. The momentary 

 effects of the forces are both infinitesimals of the second order, so that 

 a definite ratio may exist between them. Let this ratio be denoted by 

 a, so that the momentary efl'ect of the inward force may be a times as 

 great as that of the outward force. Let the inward force vary directly 

 as that power of the distance R denoted by the exponent n, and let 

 n> — 3. At a later time, the ratio of the forces will be expressed by 

 aR'/R^ = aR^"^, in which n + 3>0. Let R decrease without Umit, 

 and before it becomes equal to 0, /?"+^ will become equal to \/a, since 

 a is finite. The forces will then be equal. 



At this time, the inward \elocity will reach its maximum, since it 

 has previously been increasing from an excess of inward force, and 

 must afterwards decrease. If this velocity is finite, it must be ex- 

 tinguished, before R = 0, by the unlimited decrease of R^^. 



If w = — 3, the inward and outward forces are always equal or 

 never equal. If o>l, the inward force is always the stronger; if 

 o<l, it is always the weaker. The various results to be expected 

 from such conditions ha^'e been discussed in previous treatises, for 

 example, by Price (Treatise on Infinitesimal Calculus, Vol. III., 

 p. 487), and will not be examined in the present article. 



