Bremilcer — Errors Affecting Logarithmic Computations. 4G3 



§ 13. 



THe formulas set forth above are siifficlent to indicate the 

 probable error and the mean error of the result of any computa- 

 tion according to any formula whatever, and to show how great 

 is the likelihood that the error is contained v/ithin any assigned 

 limits. In order to show this by an example, we shall examine 

 more carefully some formulas according to which the third side 

 in a spherical triangle may be found if two sides and the in- 

 cluded angle are given. 



A. First let us consider the formulas by means of which the 

 third angle is computed by means the auxiliary angle m where 



we put 



-Qf „__ smi(7 Vsin a sin b 

 sin ^{a—b) 



and find 



<,;„ 1 ^ _ sin^ (7 i/s'in « sin ^6 _ sin|(a — 5) 

 cos /J. sin jii 



From this formula log cot m is computed by the addition 

 of three logarithms, from the sum of which one is subtracted. 

 Each one of these is assumed to be affected by an error f^ (which 

 in log sin a aad in log sin b is in fact diminished a half on ac- 

 count of the division by 2, if account is taken of the half-units 

 arising in the division of the last decimal place). So we have 

 the equation 



/ (log cot /O = /'i+i/' ,+lf\-f\ 



in which the subscripts indicate that the various quantities /' 

 are independent. 



Then according to the equations sot forth in § 12 we shall 

 Lave 



/(log cos /O = sin2,,f/',-f.i/',+i/'3-/',)+siaV/'x+/'5 

 /(log sinu) = - cosX/'i+l/'a+^Z' 5-/'4)-cos W2+/'i 

 ^(logsin^c) =/(log sin|C7|/sin a sin 6)— /(log cos//) 



= f'x+if'2-\-lf\-sm 2//( /'x+i/'2-f l/'a-/'.) 

 -sin2/i/"i-/'. 



