THE ORBITS OF THE EIGHT PRINCIPAL PLANETS. 113 



p' T =+0.0226127, log. p" =8.3543534 ; 



p r =+0.0402201, log.p r =8.6044432; 



p vi =+0.0129519, log. p" =8. 1123323; 



^"'=+0.0238090, log. ^""=8.3767411 ; 



q =+0.0844678, log. q =8.9266911; 



q 1 =+0.0149991, log. q' =8.1760651; 



q" = 0. log./ =-oo ; 



q" =+0.0214548, log. q'" =8.3315249; 



q' r =0.0035434, log. q' T =7.5494159n; 



q v =0.0165138, log. q v =8.2178478; 



q vi =+0.0039012, log. q ri =7.591 1972; 



q r " 0.0200698, log. ? r "=8.3025434rc. 



Now, adding the logarithms of m-^-na, m'-^-n'a', &c., which are given in 5, 

 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. 



log. _^_ =85.9443779 ; log. q =85.9208552 ; 



na na 



log.p'^L =86.9845985; log. q'^ =86.4020721; 



n a na 



P"^, oo log.^ =-00 



m'" m'" 



p'"^- =85.9337058; log. q" - m =85.8821396; 



p' T -^ jr =89.5793573 ; log. q"^~ v =88.774419871 ; 



p r -^ =89.4372661; log. q r -^- f =89.0506707w; 



log. p r! -"L_=88.2449047 ; log. q vi - =87.7237696 ; 



n a' 

 p-^-88. 7292393 ; log. 9 "'-^,=88.6550416n. 



7. These quantities are now to be substituted in equations (408) and (409), in 

 connection with the values of N', N", N" 1 , &c., corresponding to the different roots. 



For the root gr= 5M26112, we find, 



_ 0.612517 1.5865662 _ 14.046564 



TO" ' y ~ TO lT ^ ~T6""~ 



Whence (3=211 6' 26".8; and log. #=9.0830567. 

 Therefore, for the root g, we have the following values, 



JV =+0.121076, N' r = 0.00002517, 



N' =+0.0148671, N r =0.00003200, 



N" =+0.0106496, N rr =+0.00002804, 



N'" =+0.0021278, JV r// =+0.000000775. 



15 February, 1872. 



