80 THEORBITOFURANUS 



If we subduct the effect in question when necessary, the remainder will be the 

 effect of the secular variation of the longitude of the perihelion of Uranus, to 

 which we shall revert presently. 



Let us next introduce such a change in the eccentricity of Uranus as shall pro- 

 duce the term 34:".899 sin g, and ascertain its effect on the other terms. For this 

 purpose we must determine he by the condition 



(2 — L-)^e = 34".899 



which gives 



^e=17".464 = . 0000847. 



A change of this amount in ^e will introduce the following terms in hv and 5p 



hv = 34".899 sin rj -\- 2".048 sin 2g 

 cos T^-^p = 20 — 844 cos g — 59 cos 2>j. 



Subtracting these terms from the expressions previously found we have 



5t7 = — 0".144 sin 2ff — 6".682 cos g — 0".261 cos Qg. 

 cos'4'5p = -|- 4511 -)- 167 cosg -\- 10 cos 2g — 16 sin^-(- 1 sin 2rjr. 



Again, let us put 



ehTt = 3".342 = .0000162, 



we shall have the elliptic terms 



hv = — 6".682 cos g — 0".391 cos 2g 

 cos ^^p = — 162 sin g — 11 sin 2g. 



Subtracting tliese expressions the constant terms, independent of the mean longi- 

 tude of the disturbing planets, are reduced to 



^v=z — 0".144 sin 2j + 0".130 cos 2g. 

 cos^ap = 4511 + 167cos^ + 10cos 2^7 + 146 sin ^4- 12 sin 2(7. 

 0.43429 5p = 1969 + 73 cos^ + 4 cos 2g + 63 sing + 5 sin 2g. 



In the last equation we have introduced the constant -[-.0000008 produced in Sp 

 by the combined action of Venus, the Earth, and Mars. The effect of each planet 

 is computed by the approximate formula 



Secular Varia(!o)is. 



The following inequalities result from the secular variations of the eccentricity 

 and longitude of perihelion produced by each of the disturbing planets, T being 

 the time expressed in centuries. 



From the variation of the eccentricity 



// // /' 



Action of Jupiter, iv = —\.2\C,T »\n g —0.012Tsin2g —O.OObTsmSg 



Action of Saturn, — 9.1S2 2' sin c; —0.538 7' sin 25r — O.OSaTsinS? 



Action of Neptune, — 0.502 T sin 9 —0.030 T sin 2;/ —0. 002 T sin 83 



