THE ORB IT OP UPwVNUS. 51 



Perturhations of Latitude. 



Those are computed from the formulfe (27) and (40), no reductions being made 

 from hh and hri to h^) and hq, but the perturbations of the latitude being computed 

 directly from the former by (40). We have only to represent the expressions for 

 hk and hvi by 



hh = — S^c cos N — 2«s sin iV 

 hr, = 2a'<. cos N -{- ^a\ sin N 



and substitute u for ti in the equations (40) from which ^[3 is computed. 



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



table. The values of 



• d7i 1 , clJi 

 = cos k y 



are first formed from those terms of /;, on pages 37 and 38, which contain cr as a 

 coefficient. Tlicn, having for each original term of R 



6R m' 6h -.J 



;= cos N 



dy Ui 6y 



all the terms which have the same coefficients of ?. and a' in N are combined into 

 two depending on (/ and I' as sliown in tlie case of li on page 36. The coefficients 

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



headed . 



dy 



The value of "^^ sin N being formed for each term of E, all the terms depending 



on the same multiples of ?. and '/J are combined into two, of which the coefficients 

 are given under the proper heading. The terms of (i -{-j) 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 hk and 8yj 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 - — — ■ already given. 

 a, cos \// sni 1 



The quantities given in tlie four columns under ^k and — ^y; show the values 



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



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



