in the Secondary Systems. 495 



A 3 ( 1 — ^ e 2 ) nearly, from which we obtain 



Aoe = — ■= — nearly; 



and formula (19) becomes 



a ' 0mA8A 21 mAe 2 SA 21 mA 2 e 2 SD mB . 



BA is the change of level at the extremities of the major axis 



arising from the variation of the primary disturbance; and 



regarding these tides as conforming to dynamic principles, their 



maximum range must be proportional to the force producing 



them, multiplied by the square of its time of operation. As the 



force in these cases varies inversely as the fourth power of the 



distance, while the square of the time, according to Kepler's 



third law, is directly proportional to the cube of the same quan- 



Se 

 tity, the maximum value of BA may be represented by ^r. Now, 



if W be the angle which the satellite describes during the time 

 the tidal force requires to produce its full effects, v being the true 

 anomaly reckoned from the higher apsis, and e the relative eccen- 

 tricity of the orbit, then 



8A=-^cos(v-~W). ..... (23) 



Substituting this value for SA, and for SD its approximate value 

 D^cosv, formula (22) becomes 



^„ 9m ASe cos v — W 21 m A 2 e 2 eD , cos v , nA . 

 8F= 5D^ 10 W> >' < 24 ) 



the middle term of the second member being rejected as incon- 

 siderable, and U 1 denoting the mean distance. 



Formula (18) expresses the attractive force of the satellite on 

 the primary supposed to be a sphere; F is the extent to which 

 this force is augmented by the ellipticity of the satellite, and B¥ 

 is the periodical change in the value of F in consequence of tidal 

 fluctuations. The second term of the value of BF in equation 

 (24) would be the same if the body were entirely solid ; and 

 accordingly it could not be expected to lead to non-periodical 

 alterations in the orbit, but it has been retained to show that 

 analysis leads to the same conclusion. Now to express the effect 

 of these forces on the orbit which the satellite describes around 

 the centre of the primary, the values of F and §F must be mul- 

 tiplied by , M being the measure of the attractive energy 



