116 SECULAR VARIATIONS OF THE ELEMENTS OF 



of the node will, in this case, be equal to (jt. The analysis of § 19 being applied 

 to equation (416), will show that 



maximum tan q>=N-{- Ni-\-N 2 -{-N 3 -\- &c. ; "I . . , fi . 



and minimum tan <p=N— \ -ZVi+iVa+iVg-f- &c. \ j 



We shall now substitute the numbers which we have already computed, in these 

 equations, for the purpose of determining the maximum and minimum values of 

 the inclinations of the different orbits to the fixed ecliptic of 1850, and the mean 

 motions of the nodes of the different planets on that plane. 



9. For the planet Mercury, we have, 



Maximum tan <p = N+ N,. + iV 2 + N 3 + &c. =0.1 87242. One-half of this is 

 0.093621, which being less than N, it follows that iV exceeds the sum of all the 

 remaining terms ; consequently, the mean motion of Mercury's node is equal to g, 

 or — 5". 1261 12. The maximum inclination of his orbit to the ecliptic of 1850 is 

 10° 36' 20"; and the minimum inclination is 3° 47' 8". 



The substitution of the numbers for the other planets shows that the minimum 

 inclinations of all the other planetary orbits to the ecliptic of 1850 are equal to 

 nothing; consequently, the mean motions of the nodes on that plane are indeter- 

 minate. The maximum inclinations of the different orbits are as follows: — 



Max. inclination. Max. inclination. 



Venus, 4° 51' Jupiter, 2° 4' 



Earth, 4 41 Saturn, 2 36 



Mars, 7 28 Uranus, 2 42.5 



Neptune, 2 22.7 



Having thus given the solution of the fundamental equations for the assumed 

 masses, it now remains to determine the coefficients depending on the variation of 

 the masses. This we shall do by using the same finite variations of the masses as 

 were employed in finding the similar coefficients of the variations of the constants 

 on which the eccentricities and perihelia depend. 



10. If we now suppose that (U=-(-1.5, we shall obtain the values of the funda- 

 mental quantities which are to be used in the computation by simply making all 

 the terms of equations (153-169) positive. We shall then obtain the following 



3 1 



Fundamental Equations for u=-\-—: or for m= . 



H J [ ^2 • ' 1946300.4 



A =(/ 2 +38.201888.r/ +183.882325; 

 ^' =i f-f 23.2189305.,7-f 98.9957893 ; 

 vl"=# 2 -j-18.9824430.'(/-)- 73.7725468 ; 

 A,=<f 4-I8 .4081571.(7+ 60.3279033; 

 4,=0 2 -j-13.19285O4. /7 + 8.981549; 

 J 3 =#4-26.3821161.#-j- 9.881338; 



(419) 



