THE ORBITS OF T 11 E EIGHT PRINCIPAL PLANETS. 177 



CHAPTER V. 



TABULAR VALUES OP THE ELEMENTS OF THE PLANETARY ORBITS. 



1. IN order to complete the subject of the secular inequalities, and render the 

 preceding investigation of greater practical value, we have reduced all the astrono- 

 mical elements to tables. The tabulated values for the planets embrace a period 

 of 7200 years ; commencing 6400 years before 1850, and continuing till A. D. 2650. 

 These are given at intervals of a century during the entire period. The elements 

 of the earth's orbit, together with the precession of the equinoxes in both longitude 

 and right ascension, and the obliquity of the ecliptic to the equator, are given for 

 an interval of 16,000 years ; commencing 8000 years before, and ending 8000 years 

 after the year 1850. They are also given at intervals of a century during the first 

 half of that period, and at intervals of four centuries and eight centuries for the 

 remainder of the period. The tabulated values are computed from the constants 

 corresponding to a mass of the Earth which is equal to the assumed mass increased 



by its tenth part, or for m"= - of the sun's mass. This value of the Earth's 



335172 



mass corresponds to a solar parallax of about 8".775, which is but little less than 

 the recent determinations of that element; and it is remarkable that this value of 

 the Earth's mass is very nearly equal to that which permits the planetary orbits to 

 attain a greater eccentricity than any other mass moving at the same distance. 

 Slight changes in this value of the mass would therefore produce only very incon- 

 siderable changes in the variation of the elements of the planetary orbits. 



2. The eccentricities and places of the perihelia have been computed from the 

 values of A, 7i', h", &c., ?, I', I", &c., given by equations (C) (page 32), by substituting 

 the values of N, N', N", &c., N W, #,", &c., ft ft, ft, &c., g, ffi, g t , &c., given in 

 Art. 31 (pages 64 and 65), by means of the equations 



tan &=h+-l, tancrWi'-f-r, &r. ; e=7t-=-sin ts=?-=-cos or, e'-=-7t'-4-cx'=Z'H-cos /, &c. 



In like manner the nodes and inclinations of the orbits, on the fixed ecliptic of 

 1850, have been computed from the values of q,p, q',p', &c., given by equations 

 (F) page 1 1 1, by substituting the values of N, N', N", &c., N lt NJ, W, &c., ft ft, ft, 

 &c., g, g } , 2 , &c., given in Art. 17, pages 134, &c., by means of the equations 



tan.B=_p-r-q, tan 6'=//-H7', &c. ; tan $> =p -r-sin 9 =q -=-cos 0, 1 



tan <2>'=y-=-sin ff=q'^-cos 0', &c. j 



23 April, 1872. 



