192 



SEARLE. 



discussion, it will be sufficient to employ the Mords outward and in- 

 ward for forces tending to increase or to diminish the distance between 

 the moving particle and the central point. 



The outward force is left without a name in the usual treatises upon 

 our present subject. They make it clear, however, that such a force 

 exists, and that it varies inversely to the cube of the radius vector. 

 For instance, we find in Price's Treatise on Infinitesimal Calculus, 

 Vol. Ill, p. 462, the equation d'r/dt'^ - hyr^ = - P, in which P 

 denotes a central force, and h denotes the constant r-dd/dt resulting 

 from the law of inertia. This equation, in the form dh/df — W-jr^ — P, 

 denotes the excess of a force tending to elongate the radius vector r, 

 and inversely proportional to j-^, over any force tending to decrease 

 r. If Ji^/'fiKP, the inward force is the stronger. 



Since it will also be convenient to have an expression for tlie trans- 

 verse motion in terms of the outward force, a geometrical proof of the 

 principle just stated will here be added. 



In Fig. 1 , let C represent the fixed point about which the moving 

 particle describes its orbit, and AB the straight line through which that 

 particle moves under the law of inertia during one of the intervals 



between successive applications of force ulti- 

 mately to be supposed continuous and conse- 

 quently infinitesimal in the second degree. From 

 B let fall the perpendicular BD on CA; and at B 

 erect a perpendicular on CB, intersecting CA, 

 produced if necessary, at F. wSince CB>CD, 

 while CF > CB, there is a point E between D and 

 F such that CE = CB. When AB becomes in- 

 finitesimal with respect to CA or CB, DE = EF, 

 as is easily shown, so that DF = 2DE. The effect 

 of outward force in elongating CD is represented 

 by DE. 



The transverse motion represented by BD is 

 inversely proportional to CD, according to the 

 fundamental law of equality of areas, which need 

 not here be repeated. The squared transverse 

 motion is therefore inversely proportional to the 

 squared radius vector, which is equal to the pro- 

 duct of CD and DF. Accordingly, DF, and 

 consequently DE, is proportional to the inverse cube of CD or CA, 

 which are ultimately equal; and the transverse motion is expressed 

 by the square root of the product of the radius vector and twice the 



Fis 



