2] ON THE PERTURBATIONS OF URANUS. 19 



86-552= -57967 Sx + 0'3018 A, -1-0188 A s -5-3704 h 3 



+ 0-1460 j?! + 0-1056 2\ + O'l 149 p. t 



3725= - 1-3958 8?/ s + 0'0254 k, -0-1501 ^- 1-2407 k, 



+ 0-0316 ft + 0-0095 ft + 0'0127 q 3 



22. If in the expressions before given for Sx,, and Sy, we substitute 

 e = 0'046679 and e-sr = 50 15''8, we obtain 



Sx 2 = 0-0074608^+0-0089748^ 

 By a = -0-008974 8^ + 0-007460 Sy, 



Substituting these values in the equations (x) and (y), and in those just 

 found, it may be seen that by adding to the latter equations 



0-006768 (a;) + 0'040287 (y) 

 and -0-001869(x) + 0-008187(y) respectively, 



Sx, and 8y 1 will be eliminated, and we shall obtain the following equations : 



(1) 89-641= 0-3252 /*,- 0-9637 A s - 5-3297 / 



+ 0-0165 ^ + 0-0876 ^ + 0'1368 k, 



+ 0-0032 q, + 0-0017 (j., + 0'0436 q. 



(2) 3-695= -0-0065 h,- 0'0152 7^-0-0112 h, 

 + 0-0288 ,-0-1323 A 3 - 1-2129^ 

 - 0-0022^-0-0015^,,- 0-01 13^.., 

 + 0-0323 ^ + 0-0099 q, + 0'02l5q. i 



23. These equations would be sufficient for determining the mass of 

 the disturbing planet and its longitude at the epoch, if the eccentricity of 

 the orbit were neglected. We will now proceed to find equations from the 

 ancient observations for determining the eccentricity and longitude of the 

 perihelion. 



The equations of condition given by the ancient observations are the 

 following : 



62-6 = 8e - 0-8776 Sx, + 0'5402 8x. 2 + 0'8712 /*, + 0'5 180 h, 



- 39-31 Sn - 0-4795 Sy, + 0'8415 8y. z + 0'4909 k, + 0'8554 k, 



+ 0-0314 /i 3 -0-9999 ^-0-8640^-0-5055^, 



+ 0-9995 :j + 0-0145 q t - 0'5035 ^-0'8628 q 3 



32 



