44 RELATIONS PERTAINING SIMPLY [BOOK I. 



17 22 38&quot;.G4. To this value of E corresponds log B = 0.0000040 ; next is found 

 in parts of the radius,^ = 0.3032928, sin E= 0.2986643, whence - 2 \ E-\- ^ sin E 

 = 0.1514150, the logarithm of which = 9.1801689, and so log A* = 9.1801649. 

 Thence is derived, by means of formula [1] of the preceding article, 



2 4589614 log - 3 - 7601038 



log A* ..... 9.1801649 log^l 1 ........ 7.5404947 



log 43.56386= . . 1.6391263 log 19.98014= ..... 1.3005985. 

 19.98014 



63.54400 = *. 



If the same example is treated according to the common method, e sin E in 

 seconds is found = 59610&quot;.79 = 1633 30&quot;.79, whence the mean anomaly = 

 49 7&quot;.85 = 2947 ^ 85. And hence from 



log &( -)*= 1.6664302 



is derived t = 63.54410. The difference, which is here only i^t^nr part of a day, 

 might, by the errors concurring, easily come out three or four times greater. 

 It is further evident, that with the help of such a table for log B even the inverse 

 problem can be solved with all accuracy, E being determined by repeated trials, 

 so that the value of t calculated from it may agree with the proposed value. 

 But this operation would be very troublesome : on account of which, we will now 

 show how an auxiliary table may be much more conveniently arranged, indefinite 

 trials be altogether avoided, and the whole calculation reduced to a numerical 

 operation in the highest degree neat and expeditious, which seems to leave 

 nothing to be desired. 



- 



39. 



It is obvious that almost one half the labor which those trials would require, 

 could be saved, if there were a table so arranged that log B could be immedi 

 ately taken out with the argument A. Three operations would then remain ; 

 the first indirect, namely, the determination of A so as to satisfy the equation 



