THE ORBIT OF URANUS. 161 



* Number of 



No, Date. Equations. observations 



" iu R. A. 



4- 10.6 = 2.60 !^l + 0.03 ^p 58 



= 4- 8.2 =2.60 -fO.Oo 64 



= -j- 6.3 =2.09 +0.03 41 



= 4- 7.0 =2.60 +0.05 60 



+ 6.6 =2.59 +0.08 88 



+ 4.3 =2.09 +0.03 35 



+ 3.9 =1.57 +0.04 37 



+ 1.1 =2.60 +0.06 76 



— 1.8 =2.60 +0.02 83 



— 3.8 = 2.09 + 0.04 40 



— 9.1 =2.61 +0.02 68 



— 9.6 =2.0S4 +0.054 31 



— 10.6 =1.560 +0.040 21 

 = — 19.1 =2.600 +0.070 50 



Total number of observations in R. A., 3763. 



We have next to express the vahies of (Y/. and !'p iu terms of the corrections to 

 the elements. Differentiating the expressions 



X = / + (2« — {e') sin (/ — 7t) + ^e" sin (2^ — 27r) + -J|e' sin c' sin {31 — 3;t) + etc. 

 p = » + je- + ( — <; + |e') cos (l — n) — f e" cos (2/ — "^tt) — etc., 



with respect to /, c, and rt, and reducing the coefficients to numbers, we find 

 ^ = 1 + 0.0939 cos g + 0.0055 cos 2^ 



^ = 1.999 sin ^ + 0.1 17 sin 2/; + 0.007 sin Bj 



^ = — 2.000 COS (; — 0.1 17 cos 2j — 0.007 cos 3y 



We have here put I for the mean longitude, or 



Z = w< + £ 

 whence 



dn dl 



Also, from the expression for p 



^ = 3^ — (1 — |e') cos g — ^e cos ^ — etc. 



f'M Of on 



The values of these coefficients which depend only on g are shown in the following 

 table : 



21 May, 1873. 



