FOR THE MEAN TEMPERATURE OF A STRATUM OF SOIL. 23 



taken between -^^ and +2^- 



Now / vdx =/e ^ sin ( t-xj"^ \ dx 



2^/f 



and using 2; to denote 9 I ^ p we shall obtain as the value of ^ J vdx between 

 the given limits, the expression 



V = j-(e^ + e I sin 2 (sin ^ + cos i() 



+ — ( e^— e 1 cos z (cos t + sin if) 



Putting 



ye-\-e | sin 2i=P . ( e —e \ cos s = Q, 

 we have 



a simple harmonic function of t, which can be reduced to the form 



V = m . sin {t + a) 

 by making 



*^^ 6+ ") =1 ^^^ ^=2^2 • ^^"^^ = 2^ • Q ^^K^ ") • 



7. The values of m and a can be found by the formulae of last section ; but 

 more convenient formulae when z is small are obtained by developing in ascending 

 powers of z. For the value of m we have 



w2=ig(F + Q2)=^(/%e~'_2cos2s) 



1 /, (2g)2 (2zf , \ 1 /, (2zf (2s)4 . \ 



222^^1-2 |6 ^ jlO ^^7 



=1-^I/-^11T75^'-*-^^- 



