34 



SECULAR VARIATIONS OF THE ELEMENTS OF 



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+P) in the same equations equal 

 to z, we shall have, 



z=z sin (0H-/S) ; y=z cos (gi+fl. (137) 



Whence tan (gt+^=x-r-y. (138) 



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

 It 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 

 Venus, e' 

 The Earth, e' 



=0.2056179 



=0.00684184 



=0.01677120 



Mars, 



Jupiter, 



Saturn, 



Uranus, 



Neptune, 



e'" =0.0931324 

 e' r =0.0482388 

 e v =0.0559956 

 e" =0.0462149 

 e r "=0.00917396 



log. e =9.3130610; 

 log. e' =7.8351730; 

 log. e" =8.2245642; 

 log. e'" =8.9691008; 

 log. e" =8.6833965; 

 log. e r =8.7481539; 

 log. e ri =8.6647821; 

 log. e r "=7.9625568 ; 



cj = 75 7' 0".0; 

 d =129 28 51.7; 

 vf =100 21 41.0; 

 CT '"=333 17 47.8; 

 vs' r = 11 54 53.1; 

 cr r = 90 6 12.0; 

 *j r '=170 34 17.6; 

 tff r "= 50 16 39 .1. 



Now since h=e sin or, l=e cos or, 7t'=^ sin rs\ l=e' cos cy', &c., we shall obtain 

 the following values 



h =+0.198720, log. K = 



li =+- 00528 8, log. h' = 

 h" =+0.0164977, 

 h" =0.0418510, 

 h' r =+0.0099592, 

 h r =+0.0559955, 

 h" =+0.0075707, 

 7i r "=+0.0070561, 

 I 



=+0.052813, 

 7' =0.0043502, 

 f =0.0030164, 

 r =+0.083199, 

 I' 7 =+0.0471996, 

 l r =0.00010988, 

 r'= 0.0455906, 

 Z r "=+0.0058628, 



log. h" 

 log. h'" 

 log. h" 

 log. h v 

 log. h n 

 log. 7t r " 

 log. Z 

 log. Z' 

 log. T 

 log. T 

 log. r 

 log. 1 T 

 log. Z" 

 log. Z r// 



9.2989408 ; 



7.722.6976; 



8.2174237; 



8.6217066rc; 



7.9982241 ; 



8.7481532; 



7.8791377; 



7.8485672; 



8.7227434; 



7.6385092ra; 



7.4794898ra; 



8.9201199; 



8.6739377; 



6.0042715n; 



8.6588752; 



7.7681049. 



We have already given the logarithms of m-~-na, m'-t-ria', m"-r-n"a", &c., in 5 ; 

 and if we add them successively to log. h and log. I, log. 7t' and log. Z", &c., 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. 



