FOR THE MEAN TEMPERATURE OF A STRATUM OF SOIL. 25 



in the expression for the mean temperature, we have merely to substitute zs/n 

 for z, as will readily appear upon trial. Making this substitution, we have 



e +e J sin 3 V«, Q = ( e — e 1 cos 2\//i, 



m=y(l+^wV+^jpjr^w*2« + &c.j . . (2.) 



= l+4V^*-54^^*^^-^^^-' • , • • • (^•) 



zijn — z^Jn 



tan ( T + a I = — 1 1— ' tan z>^n 



e —6 



(I-) 



'izsjn — Iztjn 



e +1. y 1 + e , , ,. ^ 



e —1 1— e 



— 'Izi^n 



Assume tan^=e 



then tan I j + «| =tan ( j + ^ I . tan aVa, . , . (5.) 



— z/^n 



Or assume tan p=e 



then tan ( J 4- a J = sec 2p . iaxi z^/n, .... (6.) 



Also, from (§ 8.) 



1 + ^nz"^ — gQ^^^2^ — &c. 

 tan ' 



fir \ 3 3( 



(i+")- 1 , 1 



' 1 — j^»22_ 



whence 



= 1 + |?122 + |%V + &C. . . . (7.j 



2 2 



2«2^ + ^W-^Z* + &C 



tan a — 



2 2 



2+^%g2+ ,i,v + &c. 



= |ng2 + &c. (8.) 



the next term being + :^^nh^. 



A good approximation to the value of tn will generally be obtained by assuming 



tan M — nz^sJ .-^ 

 45 



and putting m=sec. w. 



11. From equations (3) and (8), we learn, that when nz- is small, the in- 



VOL. XXIII. PART I. G 



