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, sin (a + Q + A sin (20 + C.) + B t (3* + Q + B, ( 4 + Q + B n (nO + ) 



For a series of twenty-four equidistant observations, / / t,, 4, 4 t n , t u , in the 



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



A = (4 + 4 + 4 + 4 + 4 + 4) 



24 



1 2, = 0.966 (4 A. - t a + 4) + 0.866 fa - 4 - t lt + 4) + 0.707 fa - 4 - As + 4,) 

 + o. 500 fa 4 4. + 4o) +0-259 (4 4 A: -*- A 9 ) A. + 44 



12^, =0.259 (A + / 4, 4) + 0.500 fa + Ao 4 4 4) + 0.707 (4 + 4 4 3 4,) 

 + 0.866 (4 + /. - / - 4) + 0.966 (4 + /, - 4 - ') + 4 - ',. 



1 2 a, = o. 866 (/, - 4 - A + 4 + * a - 4 - / + /) +o. 500 (/, - /", - 4 + 4o + / - 'u 



- 1 + 4) - '. + A. - As + ' 



12*. =0.500 (4 + 4 - ', A, + ', + in - A. - ') + - 866 (A + A - 4 - ' + At 



+ 1* -t w - t. a ) + 4 - 4 + 4 - 4t 



1 2 3 = o. 707 (A 4 4 + A + 4 At Aa + As + 4i A 9 4i + 4) 

 4 T 4 As ~r Ae 4 i 4< 



1 2(5, = o. 707 ( A + 4 4 A + 4 + Ai A As + ^7 + A ' As) 

 + 4-' 6 + Ao-A4 + As-A 2 



i2a t = 0.500 (A 4 A + 4 + A 4 Ao + Ai + As A* A + A? + A 4> A. + A) 

 -4 + 4 4 + A, Ao + A, 4, + 4, 



12^ =0.866 (A + 4 4 4 + A + 4- Ao A, + A 3 + A* A* A, + ', + / ',. ') 



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 : 



% = l/V + *,' tan C, = a, / 6, 



* 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. 



