ORBITS RESULTING FROM ASSUMED LAWS OF MOTION. 197 



(« + 3) dR/dZ = w-dR/dZ, the ratio dR/dZ being negative, since 

 R decreases as Z increases. 



We have now to determine the corresponding ratio of forces. 



The action of a constant force upon a finite distance is proportional 

 to the square of the time allowed for such action; and a force depend- 

 ing on the exponent of that distance is to be regarded as constant 

 while we regard the distance as momentarily constant. This requires 

 the time of action to be infinitesimal, and the squared time to be 

 infinitesimal in the second order. 



Similarly, the action of a constant variation of force upon the first 

 differential of a distance is proportional to the squared time allowed 

 for its action; and such a variation of force is to be regarded as con- 

 stant while we regard that first differential as momentarily constant. 

 This requires the time of action to be infinitesimal with respect to that 

 first differential, and the squared time to be similarly infinitesimal 

 in the second order. If we consider the time of action to be in the 

 first instance infinitesimal in the second order, its square is infinitesimal 

 in the fourth order. But the relative order of the two infinitesimals 

 of time is in either case that of a quantity and its square. 



In the absence of force, dR and dZ varv uniformlv with time; and 

 while the time allowed for their variation is the same for each, their 

 ratio remains unaffected by the order of infinitesimals to which that 

 time belongs. 



In the expression for the ratio of variations of force, iv-dR/dZ, 

 the factor w- is equivalent to a ratio between the squared times 

 allowed for the variations of dR and dZ, to whate\'er order of infinitesi- 

 mals these times belong. For if the original ratio dR/dZ is multiplied 

 by such a ratio between those squared times, the product will be ir 

 times as great as before. 



The square root of that ratio of squared times will be the ratio of 

 the times hy which dR and dZ are multiplied in the corresponding 

 expression for the ratio of the forces acting upon them. This process 

 is an integration merely in the sense that the integral of an infinite 

 series of squared differentials of time is the first power of such a differ- 

 ential. The ratio dR/dZ is unaffected, as shown above, by the change 

 thus indicated in the order of infinitesimals of time. 



If, therefore, u-~dR dZ represents a ratio of variations of force, the 

 corresponding ratio of forces is wdR/dZ. 



