162 



KANSAS ACADEMY OF SCIENCE. 



If we substitute A = ~ + B, T j — 

 n 

 to the more convenient form: 



B + tti COS. (0) + a, COS. 2 (0) + - 



+ 6i sin. (0) + hi sin. 2 (0) + - 



B + Oj cos. (30°) + a., cos. 2 (30°) + . 

 + bj sin. (30°) + 6," sin. 2 (30°) + . 



ST 



ST 



?j, T., — = — = ^2> 6tc., in (4), they reduce 



a, cos. 5 (0) 

 •b. sin. 5 (0) 



\+li=Vx 



+ ag cos. 5 (30°) / . , _ / 

 .+65 sin. 5(30°) 5 "^'2 — y : 



^ . (5) 



B + ai COS. (330°) + a. cos. 2 (330°) 



.+a5C0s. 5(330°)\ , , _ / | 

 + 61 sin. (330°) + 62 sin. 2 (330°) +....+ 6., sin. 5 (330°) ; "^ '1 2 — 1/ 1 2 j 



There being twelve observation equations, and only eleven constants to be de- 

 termined, we can find their most probable values. The most probable values of the 

 constants are those which make 2i/^ a minimum, ^ being any residual error. 



The eleven normal equations, from which the values of the eleven constants are 

 found, are gotten by placing the first differential coefficient of Si/^ equal to zero. 

 These normal equations, after simplification, are as follows: 

 \2B = tl ^ 



6ai = ii — ^7 +(?3 — «9 — is + ^ii) sin. 30° + (Z, — ig — Zg + ^io) cos. 30° I 

 66i = i4 — iio + (<3— ig + ij — ill) cos.30° + ("«2— ^8+^fi— ^") sin. 30° | 

 6a, = «i +^7 — ^10 — '4 + ('''+'8+^6+^2— '3 — ^9— '5 — ^i) sin. 30° 



Qbo — {l2^li — lfi — lli+lz — 1'9—lo — lll) cos.30° I 



6a; = ii+i5+Z9 — («3+«, + «ii) y ... (6) 



, 663=i2+i6+«10 — (^4+^8+^12) I 



6a^ = ;i+Z7+i4 + fio — (^2+^8+ ^3 + ^9+^5+^1 +^6 + ^-') sin. 30° | 



664 = (^2 — ^3 + ^5 — i!g + ^3 — Jg + ii 1 — ?i 2) COS. 30° 



6a5 = Zi— i- + (— Zg+is+'fi— 'i2)cos. 30° + (Z3— Zg— i6+^ii)sin. 30° | 



665 = i4 — iio + (;ii— ig+Zg — Z-) cos.30° + (i2+^o— ^8— ^2) sin. 30° J 



The following table gives the mean monthly temperatures or values of T (com- 



ST 

 puted from Prof. Snow's meteorological record), and the values of T — - — : 



■I: 



Substituting the value of Zj, l^, I3, etc., in equations (6), and solving, we have 

 A = 52.93°, a 1 = — 25.70, a2— — 1.12, 03= — 0.47, 0^= — 0.05, 05 = — 0.43, 61= + 0.10, 

 62 = 1.02, 63=0.17, 6.1=0.25, and 65 = — 0.06. 



Substituting these values in equation (3) we have 



T=/(o) = 52.93 — 25.70 cos. G— 1.15 cos. 2 e— 0.47 cos. 3 e— 0.05 cos. 4 e— 0.43"| 

 cos. 5 e 

 + 0.10 sin. e +1.02 sin. 2 e + 0.17 sin. 3 e + 0.25 sin. 4 e + 0.01 

 sin. 5 e 



Note. — In the simplification of the normal equations we use the following results, 



easily proven true: 



S cos. OT o =0, Z sin. m e =0, 2 sin. m e cos. »n e = 0, 2 cos.- m e = , and S sin.^ 



n 360 



me = -, m taking all values from to n — 1 and e== — , n being any integer. 

 2 n 



