46 RELATIONS PERTAINING SIMPLY [BOOK I. 



auxiliary table, with which we will repeat more exactly the same calculation ; 

 most frequently, precisely the same value of B that had been found from the 

 approximate value of A will correspond to the value of A thus corrected, so that a 

 second repetition of the operation would be superfluous, those cases excepted in 

 which the value of E may have been very considerable. 



Finally, it is hardly necessary to observe that, if the approximate value of B 

 should iri any other way whatever be known from the beginning, (which may 

 always occur, when of several places to be computed, not very distant from each 

 other, some few are already obtained,) it is better to make use of this at once in 

 the first approximation : in this manner the expert computer will very often not 

 have occasion for even a single repetition. We have arrived at this most rapid 

 approximation from the fact that B differs from unity, only by a difference of the 

 fourth order, and is multiplied by a very small numerical coefficient, which advan 

 tage, as will now be perceived, was secured by the introduction of the quantities 

 E sin E, ^E-\- T V sin E, in the place of E and sin E. 



40. 



Since, for the third operation, that is, the determination of the true anomaly, 

 the angle E is not required, but the tan J E only, or rather the log tan i E, that 

 operation could be conveniently joined with the second, provided our table sup 

 plied directly the logarithm of the quantity 



which differs from unity by a quantity of the second order. &quot;We have preferred, 

 however, to arrange our table in a somewhat different manner, by which, not- 

 withstanding the small extension, we have obtained a much more convenient 

 interpolation. By writing, for the sake of brevity, T instead of the tan 2 i E, the 

 value of A, given in article 37, 



is easily changed to 



. __ T g r -f-f T s y T 4 4-jf T 5 etc. 

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