TABLES OF DIFFERENCES OF MEAN TEMPERATURES. 153 



For systematic comparison of the law of the diurnal fluctuation we present the 

 resulting hourly numbers, on the yearly average as contained in the table of differ- 

 ences, in an analytical form, making use of Bessel's periodic function — ^ 



t = A^B, sin (0 + CO + B, sin (2 + a) + B, sin (3 + G,) + etc. 



' See Bessel's paper in the Astronomisclie Nachrichten, No. 136 (May, 1828). His first publica- 

 tion on the subject is contained in the Literary Gazette of Jena, in 1814. 



See also a memoir by M. A. Bravais in " Voyages en Scandinavie, en Laponie, au Spitzberg et 

 aux Feroe, pendant les annees 1838, 1839, et 1840, Meteorologie." An extract is given by M. J. 

 Haeghens in the " Annuaire Meteorologique de la France pour 1850, p. 93. 



See also Sir J. Herschel's Article, "Meteorology" in the Encyclopedia Britannica. Reprint, p. 144. 



The general formute given in this article, when applied to the case of 24 equidistant observations 

 in a cycle, change into the following expressions, which were employed for the numerical computations : 



^ = 5V(J/i+2/. + .V3 + +2/J 



12 a, = 0.9Gfi(j/,_2/„_j/,3 + j/J+ 0.866 (j/,_2/,„-2/„ + 2/J + 0.^07(2/3-2/^-2/,, +2/.n) 

 + 0.500 {y, — y, — 2/,, + 2/,„) + 0.259 {y, — y, —y„ + j/ J — 2/,^ + j/,^ 



12 6^ = 0.259(2/, + 2/„ — 2/,3 — 2/,3) + 0.500 (?/, + ?/,„ — j/,,_2/J + 0.707 (2/, +2/, — 2/,, — 2/,J 

 + 0.866 (2/, + 2/s — 2/ic — 2/.„) + 0.966 {y, + y, —y^. — yj + y^ —y^^ 



A = V a,= -f &,^ and tan C, = ^ 



12a, = 0.866(2/^— 2/,— 2/,+ 2/11+2/,—2/n— 2/i<,+ 2/J + 0.500 (2/— 2/.. — 2/8 + 2/10+2/1. — 2/ic — 2/20 + 2/.2) 



— 2/6+2/13 — 2/is +2/04 

 12 6, = 0.5OO(2/i+2/5— 2/, — 2/1, +2/13 + 2/1,— 2/i9—2/.3)+0.866(2/, + 2/,— 2/8— 2/.„ + 2/u+2/i6 — 2/.o — 2/J 

 + 2/3 — 2/9 + 2/15—2/21 



12ffl3 = 0.70T(2/i — 2/3 — 2/5 + 2/, +2/9 —2/,i — 2/13 +2/,5 + 2/1, — 2/19 — 2/2, +2/23) 



— 2/4 + 2/8 — 2/12 + Vu — 2/20 + 2/21 

 12&3 = 0.707(2/, + 2/3 — 2/5 — 2/, +2/9 + 2/„ — 2/13 — 2/,5 + 2/i, + 2/19 — 2/21 — 2/23) 



+ 2/2—2/6 + 2/10 — 2/u + 2/is — 2/22 



12a^ = 0.500(2/i — 2/. — 2/1+2/5 + 2/, — 2/8 — 2/io + 2/„ + 2/13 — 2/,i — 2/.6 + 2/i- + 2/,9 — 2/20 — 2/22 + 2/23) 



— 2/3 + 2/6 — 2/9 + 2/,2 — 2/15 + 2/,8 — 2/21 + Vn 

 12 b, = 0.866 (2/, + 2/, — 2/4 — 2/5 +2/, +2/8 —2/10 — 2/11 + 2/i3 + 2/u — 2/i6 — 2/i, + 2/19 + 2/20 — 2/22—2/23) 

 etc. 



The values B, B^ 5,, . . and C, C, C, . . are found in a similar manner as B^ and C,. 



For 12 equidistant observations in a C3'cle, as in our bi-hourly series, we use the formulas: 



^ = tV(2/,+2/2 + 2/3 + - • +2/12) 



6«i = 0.866(?A_2/, — 2/, +2/„) + 0.500(2/, — 2/, — 2/8 + 2/,„) — 2/3+2/12 

 6 6, = 0. 500 (2/, + 2/5 — 2/, — 2/„) + 0. 866 (2/, + 2/, - 2/8 — 2/10) + 2/3 — 2/9 



6 a, = 0.500 (2/, — ?/, — 2/, + 2/, + 2/, — 2/8 — 2/10 + 2/,,) — 2/3 + 2/6 — 2/9 + 2/12 

 6 6, = 0. 866 (2/, + y, — y^ _ 2/, + 2/, + 2/8 — 2/io — 2/ii) 



6 «3 = — 2/2 + 2/1 — 2/r, + 2/3 — 2/i„ + 2/12 

 6 &3 = 2/1 — 2/3 + 2/5 — 2/, + .V9 — 2/u 



6a, = 0.500(— 2/1 — 2/, — 2/1 — .Vs- 2/, — 2/8 — 2/,o — 2/ii) + 2/3 + 2/6 + 2/9 + 2/i2 

 6 6, = 0. 8 66 (2/j — 2/., + 2/1 — 2/5 + 2/, — 2/8 + 2/,o — 2/1,) 

 etc. 



The values B, B„ B, B^ . . and 0, C, G, C.. . . are found as stated. 



The above expressions, together with others, are given in Coast Survey Report of 1862, Appendix, 

 No. 22 (with erratum in 1866 report). 

 20 February, 1875. 



