306 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



From the condition w = when z = ®, there follows KJc — Bs + Aa, 

 hence 



A i fa „ „ \ „ . a /'n 2 



€ = 



s 2 + «s + 



- j (~ n 2 — a 2 — as) er"+ £ ( - s + h ) e** J sin (mx + nt) 



If — 2 is very small, and s of the same order of magnitude as a, or even 



much larger, then for values of z that are not too large, this last equa- 

 tion becomes 



, f a m 2 , \ . , 

 6 = A \7+~a lb + aZ ) S1D ( mX + ^ 



A (' a c l G 2 , \ n / X . t 



= A \j+^ U-<F® 2 + aZ ) sm 2n \L + © 



If we put s = 0.000693, then, at an altitude of 1,000 metres, the varia- 

 tion of temperature will be half as large as at the surface of the earth. 

 With this value, and the same values of L aud @ as above, there 

 results 



£ = A (0.153 x 0.576 + 0.000125 z) sin {mx + nt) 



Hence, for a mean temperature of 273°, a barometric variation of 2.45 

 millimetres is produced by a daily variation of 10° in temperature at 

 the surface of the earth. 



VI. TRANSFORMATION OF THE DIFFERENTIAL EQUATIONS FOR SPHER- 

 ICAL COORDINATES. 



Instead of the rectilinear coordinates x, y, z, the spherical coordinates 

 (r = radius; go = polar distance; A = east longitude from adopted me- 

 ridian), are to be introduced 



x = r sin go cos A, 

 y = r sin go sin A, 

 z = r cos go. 



The equations of motion of a point on which the forces X, Y, Z are 



d 2 x 

 W 



acting along the rectilinear axes, which are X=^S, etc., are thus trans- 



formed into the following : 



d*Go drdco /(?AV 



n=r W^ 2 -dt^di- rc0S(asiUG '{ < dt) ' M 



a „ a \» d * X . o • drdx ~ daodX 



4* t Bin v df + 2 sm „-% ^+ 2 r cos go -™ 



