ORBITS RESULTIXC FROM ASSUMED LAWS OF MOTION. 



195 



value of R, the outward force is expressed by mP^'^/B?, and the 

 inward force by viR^^/W, so that ?n(P^-^— R^'^)/R^ expresses the 

 excess of outward force if positive, and of inward force if negative. 

 This expression, then, denotes the actual force momentarily existing 

 at any value of R. 



Fig. 2 illustrates the meaning of the geometrical terms above defined. 

 It represents a curve of the form which will result from an inward 

 force inversely proportional to the first power of the distance, so that 

 n = — 1. The fixed point, from which the inward force is assumed 

 to emanate, is represented by C, the point of periastron by Q, and the 



point of apastron by L. The point R is the assumed position of the 

 moving particle. The line CR represents the radius vector R; CQ 

 represents Q, and CL represents L. The position of the radius vector 

 P has not been computed for this particular figure. The lines CY, 

 RY, KY, KQ, CX, HX, HL, respectively represent the quantities 

 Y, U, Z, K, X, S, H. The angle RCQ represents v, and LCQ vq. 



IV. 



Law of variation of actual force. 



The actual force, for any value of R assumed to be momentarily 

 constant, will vary with respect to R^'^, as has just appeared. 

 When n = — 2, R^^= R, and we know that in this case dR and dY 



