16 



RELATIONS PERTAINING SIMPLY 



[BOOK I. 



the former when sin Q is greater than cos Q ; the latter when cos Q is greater than 

 sin Q. Commonly, the problems in which equations of this kind occur (such as 

 present themselves most frequently in this work), involve the condition that P 

 should be a positive quantity ; in this case, the doubt whether Q should be taken 

 between and 180, or between 180 and 360, is at once removed. But if such 

 a condition does not exist, this decision is left to our judgment. 

 We have in our example e = 0.2453162. 



9.4867632 log cos IE . . . 9.9785434ra 



0.2588593 



logvX+7) . 

 Hence 



log sin i v \Jr . 



logcosiv r 



log cos \v . . 



0.1501020. 



9.7456225 1 whence, log tan %v 9.6169771 

 0.1286454 n] %v = 15730 4r / .50 



9.9656515?* e&amp;gt;=315 123.00 



log y/r .... 0.1629939 

 logr ..... 0.3259878 



III. To these methods we add a third which is almost equally easy and expe 

 ditious, and is much to be preferred to the former if the greatest accuracy should 

 be required. Thus, ris first determined by means of equation III, and after that, 

 v by X. Below is our example treated in this manner. 



loge ..... 9.3897262 

 logcos^ . . . 9.9094637 



ecosE = 



9.2991899 

 0.1991544 



log(l 



0.4224389 

 9.9035488 



0.3259877 



log sin E .... 9.7663366 

 log \j(l ecosE) . 9.9517744 



9.8145622 

 log sin 9 . . . . 9.0920395 



log sin } (v E} . . 8.9066017w 

 l( E) = 437 33&quot;.24 

 v E =9 15 6.48 

 =316 123.02 



Formula VIII., or XI, is very convenient for verifying the calculation, par 

 ticularly if v and r have been determined by the third method. Thus ; 



