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11. — Investigation of an Expj^ession for the Mean Temfperature of a Stratum of 

 Soil, in Terms of the Time of Year. By Joseph D. Everett, M.A., Professor 

 of Mathematics, he, in King's College, Windsor, Nova Scotia.* 



(Read 3d February 1862). 



1. It is a well-known property of simple harmonic functions, that the sum of 

 any tAvo or more of them having the same period, is itself a simple harmonic 

 function having the same period as its components. The same thing must be 

 true of their mean, since this is equal to the sum divided by a constant ; and 

 it will still be true when the number of components is indefinitely great, and the 

 mean becomes an integral. 



2. Let V denote a variation of temperature which is a simple harmonic func- 

 tion of the time t, so that 



v — K sin («i + E), 



where t is expressed in arc at the rate of an entire circumference to the year. The 

 mean value of v for any assumed interval of time will be 



/ 



V dt 



taken between limits corresponding to the beginning and end of the interval. 

 Performing the operations indicated, it will be found that the expression for the 



mean temperature of an interval equal to the — th part of a year, t being the 



time for the centre of the interval, is 



. n'nr 

 Sin — 



A sm(nf + E). 



m 



3. If v in last section represent a variation of temperature at depth x, below 

 the surface of the ground, then, by the theory of underground conduction, A and 

 E are functions of x. Hence the mean value v at time t for a stratum of soil 

 will be 



/ 



V dx 



fd 



dx 



taken between limits, corresponding to the top and bottom of the stratum. The 



* See article by the Author, in the Edinburgh New Phil. Journal, vol. xvi., 1861. 

 VOL. XXIII. PART I. P 



