324 Prof. A. S. Eddington on Astronomical 



We have, therefore, to oonsider the additional perturba- 

 tions caused by the terms containing V. If u is the orbital 

 velocity of the planet, and v its total velocity, 



m = M(l + v 2 /2c 2 ) 



= M(l + ( V 2 + w 2 - 2 Vi< sin <9)/2c 2 ). 

 Hence 



dm MYu ■ 



dt = -^ cosd - e (5) 



We have here made the approximation of treating the 



orbit as circular for calculating the small perturbations ; 



that is to say, we neglect e in the terms which have the 



large denominator c 2 . Tliis is the same approximation as in 



the previous papers. 



dm 

 By (5) the term Vcos#~-t- has the variable factor cos 2 #, 



and therefore goes through its period twice in one revolution 

 of the planet. It follows from Lodge's discussion (pp. 85-86) 

 that it can only give rise to periodic perturbations which 

 would be insensible to observation. Secular perturbations, 

 which we are seeking, can only arise from terms having the 

 same period as the planet, which therefore give rise by 

 resonance to continually increasing effects. Similarly, the 



term V sin 6 -j- gives only periodic perturbations. 



Cull 



In determining the secular perturbations, the terms in (4) 

 which contain V can accordingly be dropped, and the 

 equations become identical with (2). We have seen that 

 these lead to 



d?u F m 



dd 2 + U ~~ hWM 2 ' 



and the conclusions of § 1 are valid. 



3. We have neglected the eccentricity in expressions 

 1 laving c 2 in the denominator. This means that if edtn is 

 expanded in powers of e, thus — 



edvr = a 4 a x e 4 «2 g2 + • • • > 

 our approximation gives only a . It is easily seen that 

 there must be a term a x ; the term v 2 /2c 2 in Lodge's 

 equation (2), p. 84, contains a periodic part with e in the 

 coefficient. 



The treatment of orbits of high eccentricity is much 

 simplified by the aid of a geometrical theorem. It is 

 well kno*vn that the orbital velocity can be resolved 



