APPENDIX. 317 



Besides this solution, the equation admits of three others, 



= 26 24 3&quot; 

 g=148 2 35 

 0=187 40 9 



The third must be rejected, because sin z is negative ; the second, because z is 

 greater than 8 ; the first answers to the approximation to the orbit of the earth, 

 of which we have spoken in article 142.* 



The manner of making these trials is as follows. On looking at the table of 

 sines we are led to take for a first approximation for one of the values, = 8 

 nearly, or 8 -f- x. Then we have 



log sin z ....... 9.14356 -|- 89 x 



log sin 4 z ....... 6.57424 -f 356 x 



log^csinw ...... 0.91130 



;sin(0 w o] . . . 7.48554 -+- 356 x 

 e (a o = 10 52&quot; + ftfr x 

 o) + ff = T 49 3 



0=7 59 55-f-iV** nearly =8 -f-ar. 

 For the second approximation, we make 



0=7 59 30&quot; + / ; and have 



log sin 9.1431056 -f- 150 / 



log sin 4 6.5724224 + 600 of, 



Q.csmw 0.9112987 



log sin (0 a. 0) . 7.4837211 + 600^ 

 g a, _ = 10 28&quot;.27 -f T V of nearly, 

 w -f o = 7 49 2. 00 



0=759 30. 27 + T V af = 7 59 30&quot;.30. 

 The process is the same for the other roots. 



* See article 142 of the Appendix. 



