OF URANUS AND NEPTUNE. lxv 



The forces depending on the argument PSP" cannot produce any long inequality of the 

 eccentricity and perihelion, because the effect produced in one revolution of P is destroyed in 

 the next. And in the same way the forces depending upon SPSP", &c. cannot require more 

 than one synodical revolution of P and P" for compensation. 



77- We have now considered all the long inequalities of P depending upon the first 

 powers of the eccentricities. Those which are due to the ellipticity of P"s orbit are of 

 opposite sign to the principal of those depending upon the ellipticity of the orbit of P, 

 (see Arts. 164, 65), but the latter may be expected to preponderate (as they do) both because 

 the eccentricity of P's orbit is by far the greater of the two, and because the circular force 

 depending on ZPSP is greater than that depending on PSP\ (Art. 62). We may now 

 therefore proceed to the case of the exterior planet disturbed by the interior. 



78. Since the whole of the above reasoning may be applied mutatis mutandis to the 

 explanation of the perturbations of P by P, it will not be necessary to go into the same 

 degree of detail in considering this case. We must first describe the disturbing forces on 

 the supposition that the orbits are circular. 



Let PP" (fig. 8) be the planets, and suppose, for simplicity, that P' remains fixed 

 while P revolves with an angular velocity equal to the difference between the angular 

 velocities of P and P. Then the disturbing forces acting on P* are the attraction of P in 

 the direction P'P and the attraction of P on S acting in direction parallel to PS. When 

 these forces are resolved along the tangent, the tangential force is zero when the angular 

 separation of the planets is 0°, and also when P' and S are equally distant from P ; i. e. 



at an angle cos" 1 — , or about 38°1 (at the points C and C,) ; and again vanishes when 



PSP 1 =180°. It is accelerating from A to C, then retarding from C to A', accelerating 

 from A' to d t and again retarding from Ci to conjunction; and is at a maximum at 

 points between these. The normal force acts inwards about A, and outwards about A', 

 vanishing only at two points D, £),, which are nearer to A' than to A. 



As before, each of these forces may be replaced by an infinite number of forces depending 

 on the arguments PSP', 2 PSP', 3 PSP, &c. The whole tangential and normal forces 

 however do not resemble those parts depending only on 2 PSP' in their laws of variation 

 so nearly as in the former case. 



79. We shall first consider the alteration produced in the tangential force depending 

 on 2 PSP" by the variation of P from its mean place, supposing the orbit of P to be 

 circular. Now this force is positive when the corresponding force acting on P is negative, and 

 vice versa. Therefore by precisely the same reasoning as in Art. 64 it follows that there is an 

 additional positive force acting on P' while P describes the arcs aAB, a'A'B', and an additional 

 negative force while it describes the arcs B'a, Ba. And exactly as in Art. 65 the fluctuation 

 in the distance of P from S produces an additional force acting on P 1 which is of opposite 

 sign to the corresponding force acting on P, and is therefore positive while P describes the arcs 



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