I 



A STUDY OF CORRKI.ATIONS AMONG TERKESTIUAL TEMPERATURES. 339 



through a purely fortuitous accordance of the observations with the final results. 

 Actually, therefore, I have deemed it best to use for S^ simply the mean square of the 

 actually observed deviations from the normal. 



The weights to be assigned will then be proportional to IF-^S'. In order to 

 express them in convenient units I have put approximately for the weight w = W ^ 3S^ 

 This formula has not however been without some modifications as will be seen by the 

 columns TF, t', and w. Owing to the possibility of systematic errors at anyone station 

 the stations which by the formulae would be entitled to great weight have their weights 

 slightly, diminished, and no station is allowed a greater weight than 25. 



It was found by Koppen that the fluctuation was greatest in the tropics, and 

 diminished in either direction as the latitude increased. This is what we should expect. 

 We may therefore plausibly suppose its amount at any place to be proportional to the 

 insolation, or to the cosine of the latitude. The value of this cosine to a single place 

 of decimals is given in the last line of the table. 



It now remains from all the numbers of this table to derive the most probable 

 values of .i; and ij for the equatorial regions. The values given are those derived from 

 observation in each region, without correction for latitude. Putting .»;„ and t/o for the 

 values at the equator we form from each given x and // an ecpiation of condition in the 

 form 



.T|, COS (/) = X 



2/0 COS </> = 2/ 

 The final values are 



2iM; cos (^ 



' " Sw COS" <^ 

 with a corresponding expression for y. 

 We find, from the numbers of the table, 



S?M cos (/)=+ 37.5 ••••^■o= 0°13 



2wy cos (^ = - 6.2 y„ = - 0.02 



2wcos'<^= 298 p= 0.13 



c = 9° 

 Hence, for the sun-spot fluctuation : 



At = 0°.13 cos tx.t - 0°.02 sin fit = 0°.13 cos {fit + 9°) 



The expression has been derived without any reference to the actual epochs of 

 the solar spottedness. All that we have done is to assume a period of 11.13 years in 

 the temperature, and determine what constants of a harmonic fluctuation in this 

 period will best represent the observations. It now remains to compare the epochs 

 of temperature thus derived with those of the spots. This is done in Table IV. In 



