282 Prof. Encke on the Calculation 



from two radii vectores, and the inclosed angle. This quantity 

 changes between x = and x =■ \ it from ^ to :^ tt ; and the 

 introduction of this quantity would have the additional ad- 

 vantage that, if two equations were combined, in which /3 — « 

 and /3 + «, or the other similar combinations occur, the ratio 

 of sin (|3— a) and sin (/3 + a) is very simply represented by 

 equations (9). As this advantage, however, disappears in the 

 case of the other combinations, it appears more convenient to 

 use tables for the function : 



, , , 2.r— sin2j: 



^ i^) = —A -■ 9— 



■i 0^ sin j;* 



= 2A+ T4^Bar^+ f_C^*+ Ba:« , 



123 1234.5 12 34-567 



for X = the value of this quantity is = ^ 



for j: = 90 = |, 



and for the intermediate values the changes are nearly con- 

 stant. A table for this expression has likewise been appended 

 at the end. It gives the values for every ten minutes of x ; 

 and for the convenience of the calculation the logarithms of 

 the values of the function divided by the number of minutes 

 contained in the radius have been given, or 



The value of x expressed in minutes of an arc will always 

 have to be combined with these values. 



This function being introduced, and making 



(1234) ^-^"^fj =N 

 ^ ^ tang 2?, 



(19) (12 3 4) = N, 



(1234)^^^M||i = N, 

 ^ ^ tang 2^1 ^ 



the equations (16) will assume this form : 



k{t,-t,) =N tang^l|l^vI,(/3-«) + (0 12) 

 sm z y 



^(^3-/,) = N, tang?,fc^,J,(y-«)4-(0 13) 

 (20) ^-(^3-^^) = ^'2 tang^,fc^U(y-^) + (023) 



kit,-t,) = N2COtang?,^gi^vKy.f^)-H0 14) 



k{t^-h) 



