THE ORBIT OF NEPTUNE. 



69 



In order to lessen the labor of solving these equations, they have been divided 

 into groups, with respect to the years of observation, and the difference of helio- 

 centric longitude of the earth and planet. The nineteen years of modern obser- 

 vations have been divided into seven groups, of which the first and last each 

 include two years, and each of the intermediate ones three years. Then, in each 

 group of years, the equations which pe'rtain to corresponding times of the year 

 are grouped together, and will be combined into one. 



The numbers in column P. are assumed as the " measure of precision" of the 

 residuals of each equation. These numbers were inferred from the numbers and 

 excellence of the observations on which each normal was founded, the unit of 

 precision was assumed to correspond to the probable error 1".5, and no equation 

 was allowed to have a precision exceeding 11. Hence the assumed probable error 



of each equation is 



But the residuals left after the final solution show that 



the measures of precision attached to the modern positions are too great, and 



2" 4 

 that their probable errors are really about -^-. 



Column M. gives the number by which the individual equations must be mul- 

 tiplied in order that when those of each group are added together, the precision 



P 



of their sum may be 2. It is approximately o /- n being the number of indi- 



vidual equations in the group. 



To make the solution more convenient with respect to decimals, the coefficients 

 of <5.r' will all be multiplied by 10, and those of <7t and 8k divided by 10, after 

 condensing the equations in the manner proposed. 



Thus the following twenty-nine homogeneous equations are obtained : 



