11 



Nine European observations combined furnished one *normaI place of Nep- 

 tune, for September 24, 1846. Three Washington observations, October 24, 

 and three more, November 21, completed the three observed places. Com- 

 mencing with the trial of radii vectores 33 and 34, which include Leverrier's and 

 Adams' hypothetical values, 33 was found too great, and the scale was extended 

 downwards to 32, 31, 30, and 29. 



I subjoin the table of the computed daily siderial angular motions for the 

 three intervals of 30 days from September 24, 58 days from September 24, 

 and 28 days from October 24, for this scale of assumed constant radii vectores. 

 They are the results of an approximate computation only, r is the radius 

 vector, "' is the daily siderial angular motion for the first 30 days, ^ for the 

 whole term of 58 days, "^i for the last 28 days, i" is the mean daily siderial 

 motion for r =^ a =^ semi axis major. 



34 



12.8 



16.7 



19.7 



17.90 



33 



14.6 



17.7 



20.3 



18.71 



32 



16.6 



18.8 



20.8 



19.60 



31 



19.4 



20.1 



21.2 



20.56 



30 



21.7 



21.6 



21.6 



21.58 



29 



24.1 



23.4 



22.0 



22.67 



The same analogies that led to the assumption of the constancy of the radius 

 vector, also lead to the conclusion that «- must be nearly constant. Accordingly, 

 the radius vector to be interpolated from this table was that in which \_{v — *')' + 

 {v — vif] should be a minimum. 



A slight inspection of the table shows that this value of r is very nearly 30.0 , 

 and that since for this value "' = t^ .. = ,» ^ and "/ = (" very nearly, therefore a = r 



• By means of my approximate Ephemeris, I was able to compute the value of c, or the mean of the second differences 

 of the planet's daily places, in R. A. and Dec, with a certainty of an error not exceeding 0".03. The group of observations 

 of any seven consecutive nights were reduced to the corresponding value for the 4th night by the following formula, in 

 which only the difference c of the daily motions, and not the daily motionj themselves, are employed : 



Making n = the number of observations for the nth night, 



A = (^2n, \ 



A = 4 / "' X »5 \ 



' V W3 + Ms / 



\ n^-\-n^ J 

 A.= 4(i^L^\ 



We have for the normal place on the 4th night, 



a = gi-^ Fa a, + A. (03 + », — c) + A, («, + «, — 4 c) + A, (a. + », ^ 9 c) j 



Art. 1.— 2 



