280 Prof. Encke on the Calatlaiioti 



necessary to change a more frequently than is desirable, as 



— ^ — [= 4sinjr-^ will exceed 1 when x exceeds 30°, 



This calculation may, however, be abbreviated, as the cal- 

 culation of a b may be saved by immediately substituting the 

 values of (15) in the equations (16). Combining the value of 

 a b with one of the equations (16) in which the same differences 

 of the excentric anomalies occur, — for instance, the third equa- 

 tion (15) with the first (16), — we obtain 



h(t f\ /-m oN_ (1234)tang2gtangg y2(/3 -a)-sm2(g-«) 7 1 

 U[t^ ll)-{^0 1 ^)- tang 2 1, I Ili^-"^)^ S^i^ 



consequently in every equation we shall have a function of this 

 form : , , 2 ^ — sin 2 .r 



y\^) = : — : 3 



^ ^ ' 4 sm J?' 



whose value may be given in a table. 



The function may be reduced to the following form : 



p^ (x) = -| X (1 + cotango:^)— I cotang x 

 = 4 07 + y (x cotang X— 1 ) cotang x 



We have now, if A, B, C stand for the numbers of Ber- 

 noulli, viz. 



-A- = 65 B = -^^, C = ^g-, &c., 

 these equations : 

 X cotang ^-1= _A^^^-B^.r^-C^^g^« 



cotang or = A v^x — Br-—- J^' — Cr-—- -7-, x" 



° X 12 1234 1 23456 



Having due regard to the following relations : 



B = A^. i-A'^ 

 12 5 



C- ^^ ^ AB 



D = ^ .1-AC + n^. 1b%&c. 

 12 9 1234 9 ' 



we obtain for (x cotang x—1) cotang x this series: 



and 



By this series the value of the function may be easily calcu- 

 lated for small values of x; for greater ones the finite ex- 

 pression is accurate enough for calculation. The limit of the 



conver- 



-a|^-+B tIj -' + C Tifrs " + D m^W^' 



