lx 



THE THEORY OF THE LONG INEQUALITY 



69. The perturbations of eccentricity and perihelion produced by these small addi- 

 tional tangential forces are very nearly compensated in each revolution of P. For these 

 forces are in all cases symmetrically situated in the opposite halves of P's orbit so that 

 the effect on the eccentricity about perihelion is destroyed by the equal and opposite effect 

 produced by a force of the same sign about aphelion ; and the effect on the longitude 

 of perihelion while P is approaching S is destroyed by the effect while P is receding 

 from S. And similarly the effects of the corresponding normal forces may be shewn to 

 destroy each other. But it is otherwise with the circular parts of the disturbing forces. 

 These forces are of opposite signs in the opposite halves of the orbit, and therefore 

 their effects are added together, instead of destroying each other. 



The tangential force depending on ZPSP' is negative through the arc AbA'. From a 

 point near b (since e is small) to A' this force is acting about aphelion and therefore increases 

 the eccentricity, but in the arc Ab diminishes it. Again, the tangential force is positive 

 through A'b'A, and therefore in the arc A'b' diminishes the eccentricity, and in the arc b'A 

 increases it. Also the points of maximum force, and the points at which it acts most ad- 

 vantageously, all lie in the arcs bA' , b'A, in which the eccentricity is increased. Hence in 

 this position of the point of conjunctions, the eccentricity is on the average increased in 

 every revolution of P ; and it is evident that it continues to increase while A revolves through 

 aba, and continually decreases during the next half of the revolution of A. When therefore 

 A is at a, the eccentricity is at its minimum value ; it is restored to its mean value when A 



Hence in the elliptic orbit, 



d(R) dR lr d'R __ d ldR\ lr „ dR „ , , 

 dt ad<j> ad<t> 2 * da \ad(p/ da 



— SV 

 ad<p 



of which the first term is the same as the expression in the cir- 

 cular orbit, and the remaining four are the terms from which 

 the long inequality of the mean motion arises, so far as it de- 

 pends on the ellipticity of P's orbit alone. 

 Now let 



R = -m'A i cos 2PSP'=-m'A 2 cos (2cp-2<p'); 



(PR 



. : — j-j VS<p = + SmnA 2 cos (2<f> - 2<p") e sin ft 



= -im'nA 1l e sin (2<t> -2<j>'- f3), &c, 



= — 4m'nA 2 e sin (X + to), &c. ; 



this is the term explained in Art. (64.) 



d ldR\ _ I dR 1 d'R \ _ 

 T\—n 1 Via= I--TT-+--F— r-1 vla > 

 da\ad(p) \ a 3 d<p adadtp] ' 



(A dA \ 



2m'-? -2m'— - 1 sin(2A-2d>')»a.aecoSj3, 

 a* aaa / 



= Im'nAi-m'Tia—j-?} sin (\ + w), &c, 



which is the term explained in Art. (65). 



dR . n , dA* » .. . „ 



— Ve smp = -mna -r-i cos (2<j>-2<p )e sin /3, 



dA 

 =+ \ m'na -~e sin (X + w), &c. 



this is the term due to the portion of the radial force resolved 

 along the tangent, explained in Art. (66). 



— — (5 V = + 2m' A^ sin (2<j> - 2<p') ne cos /3, 

 adcj> 



= +m'nA. l e sin(X + w)> &c. 



the term due to the fluctuation of velocity, explained in Art. 



(68). Since tt- = 1 nearly the alteration of the circular tan. 



force due to the alteration in the direction of the tangent does 

 not give rise to any term of the long inequality of the first order, 

 as explained in Art. (67). 



Collecting the terms together, we have, 



= - I 2m nA? + \mna __ I e sin (\ + vs), 

 dt \ da i 



da 

 dt' 



i -m'nMtf sin(X-i--nr), by Art. (8); 

 - 2a 2 -^—^ = + 2m'na 1 M 1 e sin (\ + rar), 



which agrees with the formula of Art (16) when regard is had 

 only to the ellipticity of P's orbit. 



It appears from the above, that the effects of the causes 1° 

 and 2° (Art. 63) preponderate over those of the other two. 



Suppose (as in Art. 64) that at the time I = 0, the planets 



are in conjunction at a, then 6 =e'= tar; 



.-. \ + vs = -(2rit-nt) = -aSA 



at the time of any conjunction ; and 



da 



— = - 2m'na 2 M l e sin (2rit - nt), 



Now while aSA varies discontinuously from 0° to 180°, 

 2n't-nt varies continuously between the same limits ; therefore 

 da , ... , 



j- is negative, as it ought to be, according to Art. (64) : so that 



the explanation agrees with the analysis in the sign of the in- 

 equality. 



