30 



By expressing the true anomaly in terms of the mean and of the ascending 

 powers of the eccentricity, a direct solution of the problem may be obtained. 

 Such a formula was communicated to me by Prof Peirce. I found, however, 

 that the method of trial and error, with an assumed value of the angle of eccen- 

 tricity f, was very convenient in practice. The partial *difFerential, 



\ d ^ J 



aiforded a ready convergence from an assumed to the true value of 4,, whence 

 rt and f were derived. 



The exact solution completed the Elliptic Elements I of Neptune, as follows: 



° I II 



7t= 48 21 2.937 l„r T. T , ,0.« 



S^=130 4 35.03|M-Eq., Jan. 1,1847. 

 i= 1 46 59.54 

 e= 0.00857741 

 ^ = 21".55448 



f = 328° 31' 56". 36 M. noon Green., Jan. 1, 1847. 

 T = 164-1^0^ Trop. years 



No. 40. The star e" Aquarii, on which the IVth normal place rests, has 

 since been found to require a correction of the value of British Association 

 Catalogue, of — 1".81 in R. A., and of — 3".00 in Dec. Applying these values, 

 this first approximation towards the Elliptic Elements furnished the follow- 

 ing comparison between theory and observation in R. A. and Dec, using Prof 

 Peirce's first values of the perturbations, s v and & r : 



Obs.— Eph. Obs.— Eph. 

 A A AD 



I May 9, 1795, +1.86 — 1.30 



III — 2.99 — 1.23 



IV — 1.00 — 0.40 

 V — 0.24 + 0.00 



This result, without the correction of the catalogue, was published in the 

 January number of Silliman's Journal for 1848. It was also communicated to 

 the American Academy of Arts and Sciences by Prof Peirce, in December, 

 1847. In the same Memoir, Prof Peirce gives the perturbations of Neptune 

 by all the planets, computed to the hundredth of a second, " including all 

 sensible terms to the cube of the eccentricities." The basis of this computation 

 was the Elliptic Elements I, last mentioned. It was the opinion of Prof 

 Peirce and myself that they were sufficiently precise to form the ground-work 

 of the full development of the theory of Neptune. It will be seen from the 



* M is the mean and v the true anomaly in 1795, May 9. The accented quantities are for August 22, 1847. 9 is the 

 angle of eccentricity. 



