SECT. 1.] THREE COMPLETE OBSERVATIONS. 209 



whence = 0.7267135. Hence are derived 



COSff 



d = 1.3625052, log e = 8.3929518 n 

 Finally, by means of formulas, article 143, are obtained, 

 logx .... 0.0913394 

 log*&quot; .... 0.5418957 n 

 log! . . . . 0.4864480 n 

 . 0.1592352 n 



152. 



The preliminary calculations being despatched in this way, we pass to the 

 first hypothesis. The interval of time (not corrected) between the second and 

 third observations is 9.971192 days, between the first and second is 11.963241. 

 The logarithms of these numbers are 0.9987471, and 1.0778489, whence 



log 6 = 9.2343285, log &&quot; = 9.3134303. 

 We will put, therefore, for the first hypothesis, 



x = log P= 0.0791018 

 y log Q= 8.5477588 



Hence we have P = 1.1997804, P -{- a = 1.5541396, P -4- d= 0.1627248 ; 



loge . . . 8.3929518 n 



log(P + a). 0.1914900 



C.log(P + rf) O .7885463w 



log tan w . . 9.3729881, whence to -f- 1316 51&quot;.89, co -f a -j- 1340 5&quot;.01. 



logQ . . . 8.5477588 



lose 2.6907847 



- 



log sin w . . 9.3612147 

 log Qc sin ( . 0.5997582 



The equation 



Qc sin w sin 4 s= sin (z 1340 5&quot;.01) 



is found after a few trials to be satisfied by the value z = 14 35 4&quot;.90, whence 

 we have log sin z = 9.4010744, log / = 0.3251340. That equation admits of three 

 other solutions besides this, namely, 



27 



