Bverailcer — Errors Affectinn Ijoc-ariilimic Coiiir/utations. 473 



If instead of the factor wliicli depends on A we put the con- 

 stant 0.13 14 we shall have 



The maxiinum vahie of tlie proi;r;ble error will here be | of a 

 second, while in the formulas considered under A and C it was 

 % of a second. Furthermore, as long as c has a value less than 

 45°, formulas A, C, and D are ailecied Avith nearly the same 

 error, but for greater values of c the last equations are evi- 

 dently to be preferred. 



These examples may sufSce to show the use of the formulas 

 proposed above. Space doc^ not permit the continuation of this 

 subject in this place, but its further study may be s«:rongl;/ urged 

 upon those who are required to make frequent use of formulas, 

 as in astronomy. In general i'c v.dil appear that a slight change 

 in a formula will sometimes bring about a very great difcerence 

 in precision, as exani})ie3 A and B show; and furthermore, that 

 the computation in which the functions of whole angles are 

 used, such as vv^as proposed in D, usually affords greater pre- 

 cision than if functions of the half angles were used, althougli 

 usually at the cost of a greater expenditure of time. The de- 

 termination of the probable error ought never to be omitted ia 

 actual practice since the following formulas : 



tan M = tan a cos C 



cos (6 — Af) 



cos c = C03 a 



cos M 



which are recommended in many mathematical books instrr'.-l of 

 those v/hich we have examined under D, ought not to be used 



