160 Kansas academy of science. 



EQUATION OF THE MEAN MONTHLY 21-YEAR TEMPERATURE CURVE 

 OF LA\YRENCE, KANSAS. 



BY E. C. MUBPHT, LAWBENCE. 



Those who have examined Part I, Vol. 12, of the transactions of the Kansas Acad- 

 emy of Science, have seen on the last page a curve called, "Curve of Mean Daily 

 Temperatures for Twenty-one Years." 



In order to show clearly the meaning of the equation that I have found, I will 

 explain briefly how this curve is gotten. 



Temperature observations have been taken at Lawrence by Professor F. H. Snow 

 three times each day for twenty-one years. The mean of the three temperatures for 

 any one day gives the daily mean temperature for that day. The sum of the daily 

 mean temperatures of any day for twenty-one years divided by twenty-one, gives 

 the daily mean temperature for that day for the twenty one years. These temper- 

 atures are platted to some scale, using time as an absissa and temperature as an 

 ordinate, giving a series of points; these points are connected by a smooth curve. 

 This curve is called curve of daily mean temperatures. 



If now we add these daily mean temperatures for any month and divide by the 

 number of days in the month, we get the mean monthly temperature of this month 

 for twenty-one years. If these temperatures be platted as above, and the points 

 connected by straight lines, we get chords of the curve whose equation I have found. 

 It is an equation, or relation between mean monthly temperature and time, which 

 has the twelve mean monthly values of temperature for values of time reckoned 

 from the middle of January to the middle ef each succeeding month. 



Temperature is a periodic function of time, the period being one year. We have 

 twelve values of this function, and the corresponding values of the time to find the 

 form of the function, or the equation of the curve. 



If T=: temperature, and < = time, we have 



T = (t)(«); or if e is any angle which varies from to 360°, as t varies from to 

 365 days, we have 



T =y(e) (1) 



The general form of a periodic function is 



y(e) = A-f tti COS. e + a2 cos. 2 e+ "1 /g) 



+ 6i sin. G -f ?>2 sin. 2e+ J 



in which A, a^, a^i bj, 63, etc, are constants. 



In this case, since there are only twelve observed temperatures, equation (2) 

 reduces to 



y(e) = A + ai cos. e +a2 cos. 2 e + Og cos. 3 e -\-a^ cos. 4 e -fag cos. 5 e ) ,gN 

 4-&1 sin. Q+hl sin. 2 Q-\-b^ sin. 3 e +64 sin. 4 G +&,s sin. 5 G ) ^ '' 



In (3) there are eleven terms and eleven constants. 



The twelve observed temperatures, or values of /(e), give twelve equations, from 

 which the values of the eleven constants are to be found. 



These twelve equations, called observation equations, are 



Ti=A + ai cos. (O)-fo, cos. 2 (0) + + a,- cos. 5 (0) 



+ 6j sin. (0) + b„"'sin. 2 (O) + • • • • + '>-, «in- 5 (0) 



T2=:A + ai cos. 30° + a„ cos. 2 (30°) + . . . . -fa, cos. 5 (30°) 



+ by sin. 30° -f h., sin. 2 (30°) + . . . . -f ^5 sin. 5 (30°) V (4) 



T, 2 = A + ai cos. (330°) -f- a. cos. 2 (330°) + . . . . +a,, cos. 5 (330°) 

 -\-b, sin. (330°) -f/j 2" sin- 2(330°) + ... +6, sin. 5 (330°) J 



