THEORBITOPURANUS. 183 



TTo representing the perihelion of 1850 = 168° 15' 6".7. We then have for the 

 eccentricity and longitude of perihelion at any epoch 



e sin (7t — TTo) = 111 



e cos (7t TTo) ^ Cg -f- hk. 



In the above terms multiplied by the time we have included the secular varia- 

 tions produced by Jupiter and Saturn. If tlie perturbations of the elements due 

 to each particular planet are required, \vc have 



Action of Jupiter, 



hh = + 5".73r; hk = — 0".6087'. 



Action of Saturn, 



Ih = + 5".56r; hk = — 4.589 r. 



Subtracting these from the above expressions all tlie remaining terms will be due 

 to the action of Neptune. The values of hi and 6» are due entirely to the action 

 of Neptune. . . 



For the sake of rigor, we may suppose the perturbations produced by each 

 planet to be multiplied by a factor representing the number by which the adopted 

 mass of the planet must be multiplied to obtain the true mass. 



It will add to the homogeneousness of the theory to express the perturbations 

 of long period, wliich are multiplied by the product of the masses of Jupiter and 

 Saturn, as perturbations of the elements. These terms, as found on page 88, are 



(i'.c.O) = — 0".55siniV6— 0".()3 cos iV^o 



4- 40 .6b sin iV, — 10 .50 cos N^ 

 (r.s.l) = + 2 .64 sin iV, + 4 .64 cos N^ 



+ 7.35sinA^, + 4. 41 cos iV^, 

 (;-.c.l) = — 4.'i3sin.V, — 3 .87 cos .V, 



+ 8 .06 sin N, — 8 .38 cos .V, 



These terms, together with the arbitrary corrections of tlie elements which have 

 been applied to make them very small at the epoch, may be replaced by the follow- 

 ing corrections to the elements: 



« = — 0".55 sin iV^,; — 0".03 cos iV^e 

 + 40 .65 sin N^ — 10 .50 cos N^ 

 4-27".27 — H".7'27: 

 hh =+2 .09 sin N,-{- 1 .94 cos N^ 

 — 2.13sini\^7+ 3.71 cos A^j 

 + 1".28. 

 a- = + 1.32siniV^, + 2.32cosJ\re 

 -f 3 .68 sin A, + 2 .21 cos A", 

 hr> =z 27 sin A^ + 104 cos A', + 76 (in units of the 7th decimal). 



The amount of the perturbations of tlie elements for every half century, from 

 the year 1000 to 2200, is given in the following table. Column (1) gives the per- 

 turbations by Neptune, Saturn, and Jupiter, computed from the expressions 



