THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 



165 



CHAPTER III. 



ON THE POSITIONS AND SECULAR VARIATIONS OF THE ORBITS WHEN REFERRED 

 TO THE INVARIABLE PLANE OP THE PLANETARY SYSTEM. 



1. We shall now refer the positions of the orbits to the invariable plane of the 

 planetary system, in order to discover whether there are any laws which control 

 their mutual positions, of a similar nature to those which we have shown to exist 

 relatively to the eccentricities and perihelia. For this purpose it is necessary to ' 

 first determine the position of the invariable plane with reference to the fixed 

 ecliptic of 1850; and we can then readily refer all the orbits to that plane. The 

 position of this plane is found by the principle, that the sum of the products, 

 formed by multiplying each planetary mass by the projection of the area described 

 by its radius vector, in a given time, is a maximum. If we put y for the inclination 

 of the invariable plane to the fixed ecliptic of 1850, and n for the longitude of its 

 ascending node on the same plane, we shall have (MScanique Celeste [1162]), 



c tan y sin II =c"; c tan y cos n =d. 

 But we have 



c =m"|/ pa (1 — e 2 ) cos (p-\-m'y /j!a! (1 — e' 2 ) cos <£>' 



-f-m"V fi"a" (1 — e"-)cos ty"-\- &c, 

 c' =m\/ )ia (1 — e 2 ) sin q> cos 6-\-m'V pa' (1 — e' 2 ) sin q> cos 0' 

 -\-m"y / fi'a" (1— e" 2 ) sin £" cos &'-\- &c, 

 c"=wil/^a (1 — e 2 ) sin <p sin 0-\-m'\/ p'a' (1 — e n ) sin <p' sin 0' 

 -\-7n"y /.i"a" (1 — e" 2 ) sin <|>" sin 0"-{- &c. 



If we denote the sun's mass by unity, we shall have 

 j.i=l-\-m, (j?=l-\-m', ju"=l+m", &c. ; 

 but we shall also have 



y [.ia=na 2 , y pa'^n'a 12 , y fj."a"=n"a" 2 , &c. 



Substituting these values in equations (528), they will become 

 c =.mna 2 y'\ — e 2 cos c£> -\-m'rid 2 y / 1 — e' 2 cos <£' 



-\-m"n"a"' 2 y 1 — e" 2 cos <p"-\- &c., 

 c , =mna 2 y / l — e 2 sin $ cos Q-\-m'n'd 2 V\ — e' 2 sin <£>' cos 0' 



-j-m"n"a" 2 y / 1 — e" 2 sin <£>" cos 0"-j- &c, 

 c"=m7ia 2 y / 1 — e 2 sin <£> sin d-\-m'n'd 2 y 1 — e' 2 sin q> sin 0' 



-\-m"n"a" 2 y l — e" 2 sin <£" sin 0' -(- &c. 



(527) 



} (528) 



> (529) 



