RECORD AND DISCUSSION OF TEMPERATURES. 31 



moans, referring to months of an average length, I have added the probable un- 

 certainty obtained by comparison with the daily means. 



January 



February . 



March 



April 



May 



June 



Annual mean from 12 average months 



For the purpose of interpolation, and for the representation of the annual varia- 

 tion, a function involving terms of the sine or cosine of multiples of an angle is 

 usually adopted. In the present case, I prefer a form of discussion which makes 

 the law of the change of the monthly temperatures analogous to that of a falling 

 body. This method was adopted by Mr. J. Wiessner, and applied to the discus- 

 sion of the Washington observations. (See p. 322 of the Annual Eeport of the 

 Regents of the Smithsonian Institution for 1857.) The annual variation may thus 

 be represented by a parabolic wave. The diurnal variation has previously been 

 represented by others by parabolic arcs. Whatever form of expression we may 

 adopt, the winter curve is so irregular, owing to the short number of observations, 

 that no continuous law can be deduced; the temperatures during this season will, 

 therefore, be treated separately. If the observations were continued for several 

 years, it is probable that the lowest temperature would fall in February, near the 

 time of sunrise; as it is, we have a slight increase of temperature during January 

 and February. 



A uniformly retarded motion is represented hy s = ct — i gf, and the condition 

 for the turning point is o = c — 0, corresponding to the middle of July, or the third 

 month, commencing with the middle of April as zero. For i! = 3, c = 3(/, and, 

 putting for convenience ^ = 2, we find c = 6 ; hence, if t = number of months after 

 the middle of April, the arguments for the several months become — 



12 3 2 10—1—2—3—2—10 



the temperature in April being the same as in June, etc. Substituting these num- 

 bers successively in the formula s = 6/! — f, we find the values {R) 0, 5, 8, 9, 8 



for the months of April, May, June — Each month furnishes an equation 



of the form T= t„, + Rp, where C = the mean temperature and i) a factor depend- 

 ing on the amphtude of the annual variation. C and j), when found for spring, 

 summer, and autumn, are found to vary, and hence an interpolation is made for 

 each month. We have next to introduce a second term to allow for a shifting of 

 the epoch. Let x be the quantity addition to the arguments 12 3 2, etc., for 

 the change in the epoch, and expressed in parts of a month, we have for— 



+ « s = 6 (0 + «) ■ 



1 + a,- 6 (1 + J^) - 



2 + a; 6 (2 + a?) ■ 



omitting terms containing the second power of x. Putting px = q, we obtain, in 

 place of the first expression for the temperature— 



