OP ARTS AND SCIENCES : SEPTEMBER 12, 1865. 



In this example the part of the formula (6) depending upon ^^ D 

 is insensible. By combining the decimal part .640 of #0 i"^ the first 

 group with the first three columns, we get the decimal part of F^ from 

 F_e to Fq. Thus .640 4- .818 + .863 + .931 = .252, only the 

 first two decimals of which need be written down ; also .640 -|- -181 

 _j_ ,359 _|_ .978 = .158, for which .16 is written ; and so on. From 

 Fq to Fq the resulting sum of the numbers is written in the line be- 

 neath the one in which the numbers are written. The two lines in 

 brackets, the first combined with the decimal part of F^ in the first 

 group, the second with that of F^ in the second group, both give Fq, 

 which is also F_ of the second group. A check for the accuracy of 

 the interpolation, as has been stated, is that these should give the same 

 result. The last two figures of the interpolated numbers being thus 

 obtained, and also their differences, as represented in the second group, 

 after the initial differences of the first group only are once obtained, 

 the remaining part of the interpolated numbers is readily filled in, as 

 represented in the first group of the preceding example. 



As a second example, let it be required to interpolate the preceding 

 Right Ascensions to fourths. Putting w ^ 8 in (7), we get from the 

 given differences 



