314 CARNEGIE INSTITUTION OF WASHINGTON. 



rates may not be obtainable before the completion of an expedition, and it is 

 then often a troublesome process requiring a large amount of time and careful 

 analysis. A table has therefore been prepared giving in a compact form the 

 corrections for rate finally deduced that may be applied to the values of 

 horizontal intensity (H) or magnetic moment (w) first computed on the basis 

 of a zero-rate for the chronometer. These corrections depend only upon r, the 

 chronometer rate, and H, as observed, in accordance with the formula 



AH =-i/r(l. 1574X10-^) (1) 



derived as follows: If T represents the time of a single oscillation and AT the 

 change in T because of chronometer-rate, we have the differential formula 

 AH/AT=-2H/T; and since AT=rr/86,400, (1) immediately results. The 

 table as prepared gives the necessary corrections for values of H from 0.02 to 

 0.40 C. G. S. and for values o" m from 100 to 1,000 for chronometer rates 

 between P and 60^ 



Computations of local mean time or azimuth from astronomical observations 

 frequently require slight revision on account of small errors in the latitude 

 which was used. A means of correcting the results directly, without recom- 

 puting with the revised latitude, is provided by the use of differential formulae 

 given in Comstock's Astronomy for Engineers, page 207, as follows: 



AA . . At , . ,_, 



-T— = — sec </> cot t; —— = —sec </> cot A (2) 



A(p A(p 



A table was prepared giving the value of each of these differential coefficients 

 for various latitudes and for hour-angles (or azimuths) differing by 5° from 

 10° to 00°, but owing to the fact that the coefficients do not change linearly 

 either with respect to latitude or hour-angle, the double interpolation involved 

 in the use of the table was troublesome and uncertain. A more convenient 

 method was obtained by using a graphical process. 



A scale of latitudes was laid off as ordinates and a scale of hour-angles (or 

 azimuths) as abscissae, and the loci of points, whose functions combined accord- 

 ing to the formulae above gave the same value for the differential coefficient, 

 were plotted. By suitably selecting the values of these coefficients, a family 

 of curves was drawn so distributed as to make the work of interpolation com- 

 paratively simple, while giving results to the necessary degree of accuracy. 



It will be noted that the two formulae above are identical, except for the 

 substitution of A for t; it follows, therefore, that one series of curves will serve 

 both for the correction to azimuth and to time or longitude. 



Attention must be given to the sign of the correction. In the formula used 

 for the computation of azimuth or time, it is convenient to consider the 

 azimuth of a body east of the meridian as negative, reckoned from the south 

 through the east, and positive when west of the meridian reckoned in the 

 reverse direction. It follows, since the sign of the correction is determined 

 by the cotangent factor, </> never exceeding 90°, that when the body is to the 

 east, the sign will be plus when A (or t) is less than 90°, and when the body is 

 west the sign will be minus for values of A (or t) less than 90°; in both cases 

 the sign changes for values of A (or t) greater than 90°. 



A new method in navigation. A method for finding any one of the three angles when given 

 the three sides of a spherical triangle or for finding the opposite side when 

 given two sides and the included angle of a spherical triangle. Henry B. 

 Hedrick. 



We have from spherical trigonometry 



cos c = cos a cos 6+sin a sin h cos C (1) 



In navigation, the complements of the sides, a, b, c, are usually given, as, 

 declination d = 90°-a, latitude L = 90°-6, and altitude /i = 90°-c. The 

 angle C included by the sides a and 6 is the hour-angle t, while the angle A 



