2] ON THE PERTURBATIONS OF URANUS. 29 



38. By means of (e) and (n) of Articles 35 and 36, eliminate 8e and 

 8n from (x) and (y), and also from the equations just found, and we have 



(x) 20272= 6-5294 c, + 5'4577 Sx, + 3'3234 h,+ 8'0291 h, 

 + 6-4240 h 3 + 1-2659 p l + 0'64:77 p,+ 5'6529j9 3 



(y) 42-61= 1-25788^ + 3'3771 Sy, + 0'3822 k,+ 2'0739 k, 

 + 3-3916 4+ 0-0836 f/, + 0'0298 </,+ 0'9513 ^, 



268-02= 11-1297 &,+ 14-4919 Sx, + 5'3967 A. + 14-4642 A a 

 + 15-4449 h,+ 2-0175 jo, + 1/0266 p,+ 9'4702j9 3 



168-85= 5-0833 8y, + 14'8824 S//, + 1-5261 A,+ 8'4906 ^, 

 + 14-6979 ^ :i + 0-3320 ^ + 0-1189 (/,+ 3-8313^. 



39. Substituting for Sx.,, Sy.,, their values in terms of 8x 1; 8y 1( we find 



6-52948^+ 5-4577 Bx,= fi'5700 Sx, + 0'0490 Sy, 



1-25788^+ 3-3771 8y,= - 0'0303 8a3, + l'2829 Sy, 



11-1297 Sot, + 14-4919 8aj 3 = H'2378 8^ + 0-1300 8y, 



5-08338^+14-88248^= - 0'1335 8^ + 5-19438^. 



Hence, if we add to the two latter equations 



-1-7106 (x)- 0-03607 (y) 



and 0-00165 (x)- 4-0487 (y} respectively, 



8^ and S?/! will be eliminated, and we shall obtain the following equations : 



(1) 80 / '28= 0-2883 7^-07295 h,- 4'4559 h a 



+ 0-0138 ^ + 0-0748 ^, + 0'1223 k 3 

 + 0-1479^ + 0-0813^ + 0-1997^ 

 + 0-0030 ^ + 0-0011 & + 0-0343 q t 



(2) 3-34= -0-0055 ^-Q'0132 ^-0'0106 h 3 



+ 0-0212 ^-0-0939 ^-0'9662 k, 

 - 0-0021 p 1 - 0-0011 p, -0-0093 p 3 

 + 0-0066 gr, + 0-0017 g s + 0'0203 q t . 



