Marine Collimating-Compass 181 



Consider separately the two parts 



- 57.3 cot A (sec h - I), and -\- m tan h cosec A 

 The first is a reduction to the sextant angle A, which converts it approximately into the 

 correspondmg horizontal angle. It may be observed that this reduction changes sign as 

 A passes from the first to the second quadrant, and referring to the above equation, it 

 may be seen that when A is less than (greater than) 90°, the reduction to apply to A in 

 order to get A must decrease (increase) the sextant angle A. The second part is a reduc- 

 tion to the measured angle A, due to the mclination of the collimator to the horizon. 

 Referring again to the above equation, we see that an elevated (depressed) scale requires 

 that the value of the sextant angle A be increased (decreased) to get A. 



Introducing these reductions in (1), we have finally the approximate working formula 



D = A,- \Ac + (r - 5.00) ?; ± [A - 57.3 cot A (sec h - 1) + m tan h cosec A][ (6) 



Here the upper sign is used for sextant in normal position, and the lower for inverted 

 position; that is, for a star to the right and left of the scale, respectively. To facilitate 

 the application of this fonnula. Tables 46 and 47 (pp. 182 and 183) have been prepared. 

 The last two terms may be obtained, one du-ectly from Table 46, the other by the aid of 

 Table 47, which contains the product of two of the factors, tan h and cosec A. 



In order to investigate the accuracy of equation (5), when terms of higher orders are 

 omitted, let it be assumed that a precision in the final result of 0?02 is sufficient, since this 

 is closer than the magnetic dechnation can be determined at sea. For values of h not 

 greater than 45°, and values of A not less than 45°, the effect of the thhd-order terms on 

 X can never be greater than 0?02, if x and m do not exceed 4?0. By reference to Table 46, 

 it will be seen that under favorable weather conditions, admitting of Sun observations at 

 low altitudes, the value of x may be restricted to much less than 4?0 by a judicious selec- 

 tion of scales. In a well-constructed instrument the mcUnation, m, of the collimators 

 should not exceed 1?0. Hence terais of the third and higher orders can usually be omitted. 



A preliminary value of the argument x is obtained from Tables 46 and 47, and values of 

 terms of the second order may then be taken out of Table 48. 



To illustrate the use of Tables 46, 47, and 48 and also the mutual dependence of 

 algebraic signs, the following hypothetical example is given. The values of m are made 

 extraordinarily large in order to introduce terms of the second order. 



m = + 2?0 - 2?0 + 2?0 -2?0 



h = 14.0 14.0 14.0 14.0 



A = 119.0 119.0 61.0 61.0 



From Table 46 + 0.972 + 0.972 -0.972 -0.972 



Factor from Table 47 multiplied by ot + . 570 - . 570 + . 570 - . 570 



From Table 48 form + 0.019 + 0.019 -0.019 -0.019 



Table48fora;=(±0?97±0?57±0?02,etc) + 0.012 .000 .000 -0.012 



A = 120.57 119.42 60.58 59.43 



Equation (2), differentiated with respect to A, gives 



14 ''in A , . 



dA = r ■■ — jdA 



cos h cos m sin A 



From this and equation (1) it is evident that, for the same values of A, the influence of an 

 error in the measured angle A has the least effect on the magnetic declination when the 

 star is low. A low altitude is a desideratum not peculiar to this method alone, but also 

 to any method of astronomically determining an azimuth from a single star. With usual 

 compass-devices consisting of mirror combinations, the error increases with the altitude 



