194 SEARLE. 



III. 



Definitions of terms and symbols here to be employed. 



The words periastron and apastron will be used in their ordinary 

 senses of the situations in which the radius vector has its least or its 

 greatest value. We have just seen that periastron will always occur 

 in the cases here to be examined, but that apastron may never be 

 attained. The straight lines drawn through the fixed point in the 

 directions of periastron and apastron will be called the axes of periastron 

 and apastron. 



The geometrical quantities having special designations in the 

 present discussion are R, radius vector; Q, least value of i?; L, 

 greatest value of R if such a value occurs; P, value of R at which the 

 inward and outward forces are equal, and the inward or outward 

 velocity is at a maximum; Y, variable distance measured from the 

 fixed point, toward periastron, to the foot of the perpendicular U, 

 let fall upon the axis of periastron from any position of the moving 

 particle; Z, variable distance measured in the same manner and 

 direction as Y, having the value not at the fixed point, but at the 

 foot of that value of U which occurs when R = L; K, greatest value 

 of Z, occurring when R = Q; X, variable distance measured from the 

 fixed point, toward apastron, to the foot of the perpendicular let fall 

 upon the axis of apastron from any position of the moA'ing particle; 

 S, variable distance measured in the same manner and direction as X, 

 having the value when R = Q; II, greatest value of S, occurring 

 when R = L; v, angle between the direction of R arid the axis of 

 periastron, having the value when R = Q; vq, the value of v when 

 R — L. Hence }' = Z + i cos vq, and X = S -{- Q cos Vq. 



Designations of algebraic quantities are n, that exponent of R which 

 shows the law of inward force under consideration, so that for the law 

 of gravitation n = — 2; m, infinitesimal of the second order, express- 

 ing the immediate effect of inward force in orbits of the same system; 

 7ti', similar infinitesimal, showing the immediate efl^ect of transverse 

 motion in increasing R; e, a constant ratio between functions of R 

 and Z, these functions to be determined for diiferent values of n, and 

 the value of e to be determined accordingly; to, an abbreviation for 

 ■\/ {n -\- 3); c', the constant cw. 



Since P denotes that value of R at which the inward and outward 

 forces are equal, viP"= m'P~^, whence m' = mP"-^^. Hence for any 



