Till: OU HIT OF URANUS. 47 



In forming the next ten lines, it will ho noticed that the value of correspond- 

 ing to any vertical column is found columns to the right. It is therefore 

 necessary to extend the line r two columns at earh end. The extension on the 

 right is, however, omitted for want of space. In performing the subtractions it 

 will be convenient to copy the r's again on the lower edge of a horizontal strip of 

 paper, and, in forming the differences .' to lay the strip above the line of v's, 

 and it' columns to the right. 



On the left of each line of differences is written the factor by which that line is 

 to he multiplied. 



The mode of formation of the A"*s is evident from the formula. 



It will he seen that the same computation which gives K u gives also /_, only 

 the latter belongs u columns to the right. 



Kadi /,, and /.-. is multiplied in succession by all the A"s which lie below it in 

 tin- same column, but the product by K* is to be written columns to the right, 

 and that by A1. columns to the left. The sum of the products in any one 



S 



column 'gives the coefficient of (/**/ + tT) in the development of 





This quantity being multiplied by ~jj- = ~ *ve have cos ^p, which only 



needs to be multiplied by sec \J/ = 1.001103 to give fy>. The units of r^p and 

 invspond to the ninth place of decimals. 



All the periodic terms are to be treated in this manner, all the series of values 

 of /.-,. and /., including the constant term, being subjected to the same process. 

 15ut, when f and i are both zero, v will be infinite. Here we simply omit the v, 

 treating it as if it were zero. We thus obtain the complete value of the terms 

 with constant coefficients in r, 2 fy and fy which are given in the following table. 

 The terms multiplied by the time are still to be computed. They arc derived from 

 (20), which may be put in the form 



rtfp = J Jf { ^Jfc? - &qjt? \ Hi. 

 This expression is computed thus: 



P. W <? V? pW -qjf? 



<> -.140783 1244.31 +175.18 



1 +.99917 + 32.78 +.99972 20.90 + 32.75 +20.89 



+ .0234 + 4.06 + -234: 1.86 + -9 + 0.04 

 \\ I- have now 



Zpjt? = + 208.02 ; MSpW = + 29689 



-2gjfc<;> = + 20.93; J If 2 jjfc? = + 2988 



r,*fy = 210 nt 



+ 2986 nt cos g + 29681 nt sin g 



+ 70 < cos 20 + 697 < sin 2jjT 



+ 2 nt cos 3# + 24 nt sin 30, 



in units of the ninth place of decimals. The value of cos ^p is obtained from 

 them by multiplying by rp 8 , exactly as in the case of the constant terms. 



