THE ORBIT OF URANUS. IJ^ 



The sum of the squares of tlie residuals is in the first case 17".94, and in the 

 second 25".41, so tliat the introduction of a and b makes a decided improvement in 

 the representation of the ohservations. 



I have not attempted a rigorous investigation of the probable error of any of 

 tlicse results for the reason that the values of the probable error deducible by the 

 metliod of least squares would, in a case like the present, be entirely untrustworthy. 

 It is, however, very desirable that we should be able to form some judgment of the 

 uncertainty of the mass of Neptune. From the last system of equations of condi- 

 tion the value of //comes out with the weight 3.13, or nearly that assigned to the 

 mean result of each five years of modern observations. Regarding these results as 

 independent, their mean error would be about 0".5, so that the probable error of fi' 

 would be 0.5, and that of fi would be .005, or about ^i^ the entire mass of 

 Neptune. A probable error derived from the original equations would have 

 been much smaller, and when, in the last equations, we allow for the systematic 

 cluiracter of the residuals, it will be larger. If we suppose the theory to be perfect, 

 I conceive we may fairly estimate the probable error of the mass of Neptune to be 

 j^Q^ of its entire amount, and its possible error two or three times greater. If there 

 is any error or imperfection in the theory, the error may be much larger. 



23 May, 1873 



