T II K (HI HIT OF URANUS. 51 



J AA* 



These are computed from the formula; (27) and (40), no reductions being made 

 from (U; and by tobp and &q, but tlie perturbations of the latitude being computed 

 directly from the former by (ID). We have only to represent the expressions for 

 (V.- and br, by 



'V.- = 2a,. cos X Sa. sin N 

 fa = 2a' c cos JV-j- 2a', sin N 



and substitute u for n in the equations (40) from which bfi is computed. 



The principal steps of the computation are shown quite fully in the following 

 table. The values of 



(ft 1 i <5A 



= cos i v 



<Y *' da 



are tir>t formed from those terms of 7*, on pages 37 and 38, which contain a as a 

 cMctficient. Then, having for each original term of R 



_ 

 dy a, dy 



all the terms which have the same coefficients of 2. and /I' in N are combined into 

 two depending on y and f as shown in the case of R on page 36. The coefficients 

 of these terms, in units of the third place of decimals, are given in the columns 



headed . 



d * th 



The value of ' sin N being formed for each term of R, all the terms depending 

 2a 



on the same multiples of X, and X' are combined into two, of which the coefficients 

 are given under the proper heading. The terms of (i +/) oh sin N being formed 

 in like manner, we have, by adding the last two expressions, all the quantities 

 which enter into the formulae (27). To integrate these equations thus forming 

 the numerical values of &k and b; we have only to multiply each term in the 

 second, third, eighth, and ninth columns of the table by the corresponding values 



of ' , for which we may use the value of v -- already given, 



a, cos I sin 1" 



The quantities given in the four columns under &k and &? show the values 

 of - - a., a,., a',., a',, corresponding to each argument. From these the 

 terms of &$ are formed by equation (40) with the modification mentioned above. 



