THE ORBIT OF URANUS. 



175 



Again, the error of the standard is not necessarily constant, l>ut may contain a 

 term proportional to the time, arising from erroneous proper motions of the 



standard Mar*. Therefore, instead of supposing the residual constant, \ve slmll 

 suppo>e it of the form + bt. Each obsened correction to the theoretical latitude 

 will then give the equation, 



sin t5$ cos u$50 + a + i 7*= /. 



To facilitate the solution of these equations they have been divided into groups, 

 each group usually comprehending three oppositions, and combined into a single 

 equation multiplied by such a factor as would make its probable error half a second. 

 The factor by which the correction of the latitude is multiplied in the equation is 

 tin- same with the coefficient of /. The year 1840.0 is taken as the epoch for b. 

 Thus we have the following: 



Treating these equations by the method of least squares, we find the normal 

 equations 



n it n > ' 



63.603$+ 6.45$50 52.10a- 1.086 = + 28.33 



6.455$ + G9.59$50 5'2.60a 14.'J5t = + 111.02 



52.105$ 52.60$50 + 133.500 + 8.956 = - 76.55 



1.085$ 14.25$50 + 8.95a + 4.496 = - 23.69 



