34 



SECULAR VARIATIONS OF THE ELEMENTS OE 



If we now put the first members of equations (134) and (135) equal to x and y 

 respectively and the coefficients of sin or cos (gt-\-(3) in the same equations equal 

 to 2, we shall have, 



x=zsm(gt-\-(3); y=z cos (gt-\-(3). (137) 



Whence 



tan (gt-{-(3)=x-i-y. 



(138) 



17. In order to find the values of a; and y, it is necessary to know the values of 

 ft and I. "We shall therefore suppose that at the epoch of 1850, the eccentricities 

 and places of the perihelia of the eight principal planets have the following values, 



Mercury, 



e =0.2056179 



log. 



e =9.3130610, 



gj = 75° 7' 0".0 



Venus, 



e' =0.00684184 



log. 



e' =7.8351730 



ts' =129 28 51 .7 



The Earth 



e" =0.01677120 



log. 



e" =8.2245642 



; xs" =100 21 41 .0 



Mars, 



e'" =0.0931324 



log. 



e'" =8.9691008 



ct'" =333 17 47 .8 



Jupiter, 



e tv =0.0482388 



log. 



^=8.6833965 



s JF = 11 54 53.1 



Saturn, 



e r =0.0559956 



log. 



e r =8.7481539 



xs r = 90 6 12.0 



Uranus, 



e ri =0.0462149 



log. 



e TI =8.6647821 



© F/ =170 34 17.6 



Neptune, 



e F ==0.00917396 



log. 



e FJ =7.9625568 



CT r/ = 50 16 39 .1 



Now since h=e sin ts, Z=e cos cJ, 7i'—e' sin cr', Z=e' cos ts', &c, we shall obtain 

 the following values 



h =+0.198720, 

 7i' =+0.0052808, 

 h" =+0.0164977, 

 ft!" =—0.0418510, 

 h 17 =+0.0099592, 

 h T =+0.0559955, 

 h TI =+0.0075707, 

 7i FJ =+0.0070561, 



I =+0.052813, 



I' =—0.0043502, 



I" =—0.0030164, 



X" =+0.083199, 



l IT =+0.0471996, 



l v =—0.00010988, 



Z FJ =— 0.0455906, 



Z F "=+0.0058628, 



log. h 

 log. ft! 

 log. h'' 

 log. ft' 

 log. K 

 log. h } 

 log. K 

 log. K 



log. Z 

 log. I 

 log. Z" 



log. r 

 log. r 

 log. r 

 log. r 

 log. r 



=9.2989408: 



=7.7226976: 



=8.2174237: 



=8.6217066ra; 



=7.9982241: 



=8.7481532; 



=7.8791377: 



=7.8485672: 



=8.7227434: 



=7.6385092w; 



=7.4794898rc; 



,8.9201199; 



=8.6739377; 



=6.0042715rc; 



=8.6588752^; 



=7.7681049. 



We have already given the logarithms of m-^na, m'-^-n'd, m"-±-n"a", Sec, in § 5 ; 

 and if we add them successively to log. h and log. I, log. ft! and log. T, Sec., we 

 shall obtain the following values of the logarithms of the constants, for the given 

 epoch, which enter into the values of x and y. 



