INTRODUCTION. 



§ 1. The first important step, in the reduction of the planetary perturbations to 

 numbers, is the determination of those coefficients in the perturbative function 

 which depend upon the ratios of the mean distances. 



This work has been done by different astronomers, but lastly and most completely 

 by Leverrier, whose results were published in LiouvilWs Journal for 1841. Since 

 that time Neptune and thirty-eight Asteroids have been added to the catalogue of 

 planets. 



If, through the ordinary forms of development, we wish to determine the secular 

 and periodic inequalities of the elements of the orbits of the fifty planets, it will 

 be necessary to find the coefficients which correspond to three hundred and sixty- 

 four values of the ratio of the mean distances. 



This estimate is upon the supposition that the mutual action of the Asteroids is 

 neglected. 



2. At Professor Peirce's suggestion, and with the approval of Commander 

 Davis, the Superintendent of the American Ephemeris and Nautical Almanac, I 

 have undertaken this work, as a part of the systematic labor of a thorough revision 

 of most of the planetary theories, now being carried on, under the superintendence 

 of Commander Davis, as fast as can be done consistently with the demands which 

 the regular issues of the Almanac make upon the annual appropriations made by 

 Congress for its support. 



3. The following notation and forms are found in any work upon the theory 

 of the planetary perturbations. 



(1 — 2 a cos I + a 2 )-" = | J<;» + J™ cos I + V? cos 2 I + ¥* cos 3 Z -| &« cos i I. 



(1)6 *~ 2 1 • 2 • 3 • 4 • • • • i a V 1 + UH r D a+ l-8-(» + l)(i + 2) a+ ) 



In these formulas a denotes the ratio, taken less than unity, of the mean dis- 

 tances of two planets ; and s = •}, f , J- , &c. 



The series for computing the values of Vf, ¥]\ Vf, &c., ttf, ¥\\ Vf, &c, ¥f, ¥]\ ttf, 

 &c, or as many of them as are needed, are readily obtained from (1) by substituting 

 the proper values of s and i. 



The values of a D„ U'J, a 2 Dl b {i J, a 3 D\ b (i J, &c, are also needed ; and the series for 

 computing them are found by taking the successive derivatives of (1) and multi- 

 plying them by a, a 2 , a 3 , &c. 



