H DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



Denoting the gravity potential by <, and the dynamic heights and depths respec- 

 tively by //and D, we have the relations 



(a) 4> = ioJ/ <f> = 10D 



by which we return from the technical unit, the dynamic meter, to the m.t.s. unit, 

 the dynamic decimeter. 



ii. Fundamental Formulae for the Gravity Potential. The difference of 

 potential between any two points can be found if we know the value of the accel- 

 eration of gravity everywhere along a curve 5 leading from the one point to the 

 other. Let g s be the component of the acceleration of gravity in the direction 

 tangential to the curve s. The work per unit-mass performed against the action 

 of gravity, when a mass is displaced the length ds along the curve is then g s ds. 

 That is, the elementary difference of potential between the end-points of the line 

 element ds is g s ds, and the finite difference of potential <f> 2 <f> t between any two 

 points joined by the curve s is found by the integration 



() 4> 2 - 4>i J g. ds 



If the curve s coincides with the plumb-line, the acceleration of gravity will 

 always come in with its full value g. It the lengths measured along the plumb-line 

 be denoted by z, and the heights of the points i and 2 above sea-level by z x and z 2 , 

 the expression (a) takes the form 



(*) * - *, = - ("gdz 



Jz, 



If from (b) we pass to dynamic heights in the atmosphere, expressed in dynamic 

 meters, we have 



W H.-H^-^rgdz 



Correspondingly for the difference ot dynamic depths in the sea we have 



These formulae serve to calculate the dynamic value of given geometric differ- 

 ences of height or depth. 



12. Normal Relation between Geometric and Dynamic Heights. Introducing 

 the value (a), section 7, of the acceleration of gravity g in the integral 1 1 (c), and 

 integrating from the initial height z to any height z, we get for the corresponding 

 difference of dynamic height 



(a) H- H l = ^ > (z- r x ) - o.ooooooi543(* 2 - z?) 



By this formula we find the dynamic difference of height corresponding to any 

 given geometric difference of height. It is to be noted that in the first approxima- 



