492 Mr. D. Vaughan on Static and Dynamic Stability 



— r — in the last equation must receive an increment, the prin- 



A 

 cipal term of which will be C cos (2L — 2w), or 



C(cos2Lcos2wf sin 2L sin 2u>), ... (7) 

 in which w represents the increase of longitude during the in- 

 terval between the times of high tides at any locality, and of the 

 maximum intensity of tidal force. Denoting by N and N' the 

 sine and cosine of w, which is constant, formula (6) becomes 



dm 3M/A-C . 2T . OT , CN' . n . f 



^- = D^l-A- Sm Ism2L ^T Sm Ism4L 



CN sin 2 1 cos 4L C sin 2 IN \ , Q , 



2 + —2— ) (8) 



If this equation be integrated, all the resulting terms of the 

 second member will be periodical except the last, which will 

 express the slow permanent diminution of I ; but the term will 

 disappear when w is exactly 90 degrees, as it should be if the 

 oscillations on which the change of form depended were effected 

 without any loss of force. 



Although the influence of distant bodies in changing the plane 

 of the orbit may prevent I from sinking to zero, yet we must 

 recognize the tendency to the peculiar arrangement which reduces 

 to the lowest scale the dynamic effects of the disturbing force on 

 their surfaces of secondary planets. But though their times of 

 rotation and the position of their axes may be adjusted for attain- 

 ing this object, the eccentricity of the orbit would bring tidal 

 action into existence; and any commotions which this might 

 occasion in their seas must be attended with secular changes in 

 the size and form of their orbits. This will appear evident 

 when we consider that these tides could not reach their higheat 

 level on the parts of the satellite in conjunction with the primary, 

 until some time after the disturbing force which produced them 

 attained its greatest intensity ; and the subordinate world would 

 thus present a greater deviation from a true sphere, in passing 

 from the lower to the higher apsis, than in returning to the 

 former point. It would accordingly feel the restraint of the 

 centripetal force more intensely when retiring from the primary 

 than when approaching him ; and its motion would be retarded 

 during the former period to an extent slightly greater than 

 that to which it is accelerated during the latter. We may 

 therefore reasonably expect a secular alteration in its mean 

 motion and the size of its orbit; but it may be advisable to 

 show by analytical investigations, that such changes take place 

 on a scale corresponding to the waste of tidal power. 



Although this may be done without any hypothesis in regard 



