MAGNETIC OBSERVATIONS 289 



The analytical expressions representing the diurnal variation in magnetic declination for 

 the four- week periods from October 4, 1903, to July i, 1904, and for the interpolated months 

 of July, August, and September have been deduced from Bessel's periodic function.* This 

 function is represented in the general case by the following formula : 



D = A+B,sm {e + Q) + ^, sin (2<' + Q) +B,(3e+ Q) + B.^e + Q) + . . . . B„(ne + C„) 



For a series of twenty- four equidistant observations, i^, i„ i„ it, (^ i,^, t^^, in the 



cycle the numerical computations may be made directly from the following equations : 



A=^(t, + t, + t, + t, + t,-^ ^«*) 



1 2a. = 0.966 (/, — t,, - A, + 4) + o. 866 {t, - 1,, - /., -f 4) + o. 707 (4 -t,- 4 -1- 4) 

 -f 0.500 (/, - 4 - 4 + 4) +0. 259 (4 — t,— i„ 4- i,,) - t,, -f (,, 



1 26, = o. 259 (/i + i,, - /„ - 4) + o. 500 (/, + 4 - i,, - 4) + o. 707 (^3 + 4 ^ i,, - /„) 

 + 0.866 (/, + i,- 4 - 4) + 0.966 (4 +i,- t„ - /■„) + ^, - t,, 



12a, = 0.866 (A _/,_/, + ^„ 4- /„ - t„ - 4 + 4) +0.500 {t, -t,-t, + 4 + t,, - /„ 



4 "r '22 / ^6 ~r" M2 MS ~r Hi 

 123, = 0.500 it, + i,-i, — t,, + 4 + t„ - 4 - 4) + 0.866 (/, + ;, - /a - 4 -)- A* 



I '16 *20 '22J I *3 '9 T ^15 ^21 



I2«3 =0.707 (/, - /, - ^, + if, + ^, _ /^, _ /^^ + /^^ + /,, _ /^^ _ i^^ + /^J 

 »4 + 'a '12 + *IS '20 T" 4 



12*, = 0.707 (/, + i, — t,-i, + i,+ t,, - t,, - t,, + A, + ^9 - ^2, - ^23) 

 "> '2 *6 ~r '10 M4 1" '18 '22 



i2«, = 0.500 {t, — t^ — t, + 4 + A —4 — 4 + 4 + in — t,,~t,, + 4 + 4 — 4 — 4 + 4) 



— 4+4 4 + ^2 As + ^8 hi + ^4 



12*, =0.866 (/, + t, — t—t^ + 4+4- 4— 4,, + 4 + ii. — in—t,, + 4 + 4—4 — 4) 



and other expressions of like character for terms of higher order. For most practical pur- 

 poses the series is not in general improved by the addition of terms beyond the fourth. On 

 the solution of these equations by substitution of the observed values the term-coefficients and 

 angles may be determined by the following : 



B, = Va^ -\- bi tan Q = aj b. 



B^ = |/a./ + b^ tan Q = a,l b. 



B,= i/ a," 4- b,' tan Q = aJ b, 



B^ = \/ a^ + b^ tan Q = atJ b^ 



* First published by Bessel in the Literary Gazette of Jena in 1814; also published in his paper in 

 Astronomische Nachrichten, No. 136, May, 1828. See also Bravais' memoir referred to above and C. A. 

 Schott in Appendix No. 8 of the Report of the Superintendent of the United States Coast and Geodetic 

 Survey for 1890. 



