OF URANUS AND NEPTUNE. 



lxiii 



force acts outwards through the arcs ABd and A'B'a, and inwards in the arcs Aa, A'a, 

 and therefore produces a long inequality of epoch of an opposite sign to the former. 



73. The tangential forces produce but little effect on the epoch. Suppose an accele- 

 rating tangential force to act during the whole of a revolution. Then the eccentricity 

 is increased throughout the arc tal (fig. 7), and diminished in Idl'. The perihelion advances 

 in the arc aid, and recedes in the arc a'l'a. Therefore in the arc ab the equation of the 

 centre in increased by increase of eccentricity, but diminished by advance of perihelion, so 

 that the effects of these changes on the epoch tend to neutralize each other. And the same 

 tendency towards compensation may be shewn to exist in the other parts of the orbit. The 

 analytical expression however shews that the compensation is not exact on account of the 

 orbit having an eccentricity independent of perturbation ; and that the circular tangential 

 force produces a part of the long inequality of epoch. The circular normal force also 

 produces a portion of the long inequality for the same reason*. 



74. If we neglect powers of the eccentricity above the first, the disturbing forces 

 depending on the arguments PSP, SPSP , &c. cannot give rise to any portion of the long 

 inequality when the orbit of P alone is considered to be elliptical. For the forces which 

 produce the long inequality of the mean motion investigated in Arts. 64- — 68, arise from 

 the fact that a portion of the circular disturbing forces goes through its changes in nearly 

 the same time as the fluctuations in the motion of the planet due to the ellipticity of its 

 orbit ; that is, nearly in the periodic time of the planet. And it is because the period of 

 the force is not exactly equal to the periodic time of the planet, that the inequality is at 

 length compensated : for by this means the force and the irregularity in the motion of the 

 planet are in the lapse of time presented to each other in opposite phases, so that the 

 additional force becomes of an opposite character. And this compensation will be effected 

 more rapidly for those forces which depend upon the angles PSP', SPSP , &c, the periods 

 of which differ very greatly from that of P. Thus, the period of the force depending on 

 PSP" is nearly equal to twice that of P ; so that this force is of opposite signs in succes- 

 sive revolutions of P; and therefore the additional force in one revolution is of opposite 



* Let H, H' (fig. 7), be two consecutive positions of the 

 upper focus under the action of a normal force directed out- 

 wards; then 



2ade = HH sin PHS = HH sin /3, neglecting e 2 , &c. 

 2aed-a = - HH' cos PHS = - HH cos ft 



9 = n<+e + 2e sin/3 +2e'sin2/3; 

 .-. = de (1 + 2e cos /3) + 2de sin /? - 2ed-a cos /3 



+ - ede sin S 



■5« ! yiircos2/3; 



,-. d f = 



5 5 



2*rfw(cos/?f~ecos20)-2^(8in/3 + -esin2/3) 



4 4 



"" 1 + 2e cos /3 



IIH ,\+- 4 ecosf} 



HH 



n 



4«=°s0), 



a 1 + 2e cos /? 

 HH' is proportional to the disturbing force; hence both the 

 elliptical and circular forces produce long inequalities of epoch. 

 Let HH" (fig. 7), be consecutive positions of the upper 

 focus under the action of a retarding tangential force ; then 



2eda + lade = - HH" cos j3 ; 



.-. 2ade = - HH" (cos /3 -2e), 

 2aed-BS - - HH" sin /3 ; 



3 HH'esmB 3 HH" . . 



•■• de= -4«(l+2 e cos,J) = -4 — eSm ' J ' 

 HH" is proportional to force x velocity ; hence the circular 

 tan. force produces a part of the long inequality. 



