lg ON THE PERTURBATIONS OF URANUS. [2 



19. By means of (n) eliminate 8n from each of the other equations, 

 and we have 



(y) 42 /r 61 = 1-2578 8ft + 3'3771 Sft + 0'4100 k, + 2'1748 k, 



+ 3-3962 k,+ 0-0785 ft + 0"0433 ft+ 1"0833 ft 



(k,) 13-90= 0-4100 8ft + 1-0787 Sft + 0-1 346 , + 0-7057^ 



+ T0871 k,+ 0-0288 ft + 0'0150 ft + 0'3534 ft 



(*,) 73-38= 2-1748 8ft + 5 '9822 Sft + 07057 A, + 37767 A, 



+ 5-9998 k,+ 0-1298 ft + 0'0732 ft + 1'8715 ft 



111-62= 3-3962 S//. + 10-3315 Sft+ 1-0871 ^ + 5-9998 & 2 



+ 10-2473 4 + 0-1930 ft + O'l 110 ft + 2'9145 ft 



1-4-2= 0-0433 Sft+ O'llOO Sft+ Q'0150 k,+ 0'0732 k, 



+ 0-1 110 4 + 0-0055 ft + 0-0023 ft + 0-0375 ft 



36-72= 1-0833 8ft + 2'8969 Sft + 0'3534 k\ + 1'8715 k, 



+ 2-9145 k,+ 0-0684 f/, + 0'0375 ry., + 0'9330 q a 



20. Again, eliminating 8ft by means of (ft) we find 



(,) 0-009= -0-0221 Sft + 0-0010 ^-0'0032 ^-0'0200 k 3 



+ 0-0032 */, + 0-0009 ft + 0-0003 ft 



(As) -0-301= 0-1430 Sft-0'0032 ^ + 0-0162 ^ + 0-1274^ 



-0-0059 ft -0-001 7 ft- 0-0016 ft 



(k 3 ) -3-443= 1-2129 Sft-0'0200 ^ + 0-1274^+ 1-0769^ 



-0-0189 ft -0-0059 ft -0-0105 ft 



(ft) -0-045= -0-0062 8ft + 0-0009 ^-0-0017^-0-0059*, 



+ 0-0028 ft + 0-0008 ft + 0-0002 ft 



(ft) + 0-017 = - 0-0116 8ft + 0-0003 k, - O'OOIG k, - O'OIOS k 3 



+ 0-0008 ft + 0-0002 ft + 0-0000 ft 



21. From the equations remaining in the two groups after the elimi- 

 nation of Se, 8n, 8x 1; 8ft, it will be easy, when approximate values of the 

 mass and mean longitude of the disturbing planet have been found, to deduce 

 the final equations for determining these quantities more accurately by the 

 method of least squares. 



It may be observed, however, that the equations in each group are 

 very nearly identical with each other, and therefore two final equations may 

 be formed by simply adding together the several equations of each group, 

 after giving the unknown quantities the same sign in them all. Thus we find 



