GRAVITY AND GRAVITY POTENTIAL. 15 



tion we can neglect the term containing {z 2 z 2 ), and instead of that use the 

 approximate value 9.80 for the acceleration of gravity at sea-level. This gives 

 the approximate relations 



(a') H H x = 0.98(2 z x ), or counted from sea-level, H= 0.980 



(b') z z x = i.02(H H x ), or counting from sea-level, z = 1.02H 



That is, the number expressing a height in dynamic meters is approximately 2 per 

 cent smaller than the number expressing it in meters. 



Supposing that the dynamic height be given, while the corresponding value of 

 the geometric height should be found, we have to solve equation (a) with respect 

 to z z x . To do this conveniently we first substitute from (b') the approximate 

 values of z and z 1 in the correction term of equation {a), which is thus made linear 

 in z z x . Solving and simplifying the correction term by the introduction of the 

 approximate value 9.80 for the acceleration of gravity g u we get the equation 



{!>) z-z x = (H-H x ) + o.ooooooi637(/P- H*) 



O 1 



by which the geometrical value of a given dynamic height can be calculated. 



In practical application it will generally be most convenient to have all heights 

 measured from sea-level. We then have z x = o, H x = o, -, == g<>, and formulae (a) 

 and (b) take the form 



{a") H = z 0.00000015432 2 



(b") z= H + 0.0000001637H 2 



In order to tabulate conveniently these formulae, we shall write them in a slightly 

 modified form. In both the main term depends upon two variables, namely, g- and 

 z or g- and h, respectively. But, thanks to the small variations of g~ , we can ac- 

 count for the influence of the variations of this quantity in a correction term, while 

 the main term is made to depend upon one variable only. To attain this we shall 

 write 



w *~9.*>( I+ ^) 



The fraction contained within the parentheses will have a value never exceeding 

 0.004. Neglecting squares of this quantity as well as products of it by quantities of 

 its own order of magnitude, we bring the formulae (a") and (b") to the forms 



('") H= {0.982 0.00000015432 2 } 4- o.i(g' u 9.80)^ 



(b"') z= {i.02o 4 o8Zf + 0.0000001637ZP} ~ g ~ q 9 6o H 



The expressions inclosed within parentheses depend upon one variable only. Their 

 values are given in tables 3 m and 5 m of Meteorological Tables. They give the 

 relation between geometric and dynamic height for places where the acceleration 



