SECT. 3.] PLACES is CEBIT. 119 



and 



or 



sb gr |(2gr sin2gr)(l f si 

 ^(2 9 an2g) 



The numerator of this expression is a quantity of the seventh order, the denomi 

 nator of the third order, and , therefore, of the fourth order, if g is regarded as 

 a quantity of the first order, and x as of the second order. Hence it is inferred 

 that this formula is not suited to the exact numerical computation of when g 

 does not denote a very considerable angle: then the following formulas are 

 conveniently used for this purpose, which differ from each other in the changed 

 order of the numerators in the fractional coefficients, and the first of which is 

 derived without difficulty from the assumed value of a? .* 



1 



i 



1 etc., 

 or, 



5 ^ T~Q 7 



1 etc. 



In the third table annexed to this work are found, for all values of x from 

 to 0.3, and for every thousandth, corresponding values of computed to 

 seven places of decimals. This table shows at first sight the smallness of for 



* The derivation of the latter supposes some less obvious transformations, to be explained on another 

 occasion. 



