291 



r = ^ + cf. .... ... (29) 



We have now, remembering that linally we will put 6=:0. 



By the aid of (10) and (14) to (16) we find easily 



^-^dr=?^{2dB-du) = ?^dii (30) 



Here an angle (x has been introduced, of which the geometrical 

 meaning is easily seen. If we take polar co-ordinates tj and (/ with 

 the second (empty) focus as origin, then ii bears the same relation 

 to rp as the mean bears to the true anomaly. Therefore, since 



r' df=: a' \^l—e'dM, 

 the equation connecting (f and (.i is similarly 



q' dff =: a' \/\ — e^din. 

 We have the formulas 



fi — = E -j- e 8171 1) 



Q COS (f = a {cos E 4- g) ^ = rt (1 + g COS e) . (31) 



/ . ail-e-") 



Q sin f£=:a V\ — e^ sin e o z= . 



^ ^ 1 — e cos (p 



The angle n is easily seen to be proportional to the "action", if 



for the mass we take x'. In that case the components of the momen- 



dxi 

 turn become vi = ^"^ —-, and 

 dt 



j'2Tdu = crn 



I now take the fourth parameter :; as variable. We then have 

 Here r = 2a — q must be expressed as a function of the elements 



by (31). 



AM is of the order of the perturbing masses. If aS=0 the motion 

 is Keplerian : M, G, fJ, g, 0^ are constants. 



For use with the elements IV, for which I will not try to coin 

 a name, a development of the pei-turbative function S according to 

 the trigonometric functions of multiples of (i would be required. 

 This can be derived from the well known development in function 

 of the mean anomaly by substituting q for r, (p for v, — e for e 

 and fi for /. 



19* 



