306 THE MECHANICS OF THE EARTh's ATMOSPHERE. 



From the conditiou ic = () when .c = 0, there follows /iA- = -Bs + A«, 

 heuce 



h 

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



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

 tion becomes 



£ = A 





= ^ [J+T, l?^<fW + ^V '''' ^'' ^ X + © 



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

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

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

 results 



f = A (0.153 X 0.576 + 0.000125^) siu {mx + ni) 



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; oj = polar distance; A = east longitude from adopted me- 

 ridiau), are to be introduced 



x= r sin go cos A, 

 y = r sin co siu A, 

 z = r cos CO. 



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



(Px 

 acting along the rectilinear axes, which are A'=|y2, etc., are thus trans- 

 formed into the following: 



dt' \dtj \dt J 



. d^X „ . drdX dojdX 



A= r sin 0.^^,-f 2 sm o.-^^ ^^-f 2 r cos a. ^-^^ 



