1 6 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



of gravity at the sea-level height z x has the special value 9.80. The last term in 

 each equation gives the correction for other values of g . The value of this cor- 

 rection is given in tables 4M and 6 m of Meteorological Tables. These tables can 

 thus be used to pass from geometric to dynamic heights and vice versa, the only 

 supposition being that we know the value g of the acceleration of gravity at sea- 

 level, which is found either by table 2 m, or by reduction to sea-level of the value 

 of the acceleration of gravity found by direct determinations at the earth's surface. 

 Proceeding in this way, we find the dynamic heights above sea-level both of the 

 ground and of points in the free atmosphere. The height of the ground will con- 

 tain an uncertainty due to that of the reduction of g to sea-level. But the heights 

 of the points in the free atmosphere above the ground will contain no error due to 

 this reduction. 



13. Normal Relation between Geometric and Dynamic Depths. Introducing 

 the value (b), section 9, of the acceleration of gravity below the integral sign of (a) 

 section 11, and integrating from sea-level, where D = z = o to any depth z, we find 

 the corresponding value of the dynamic depth D 



or 



(a) Z>= z + 0.000000 1 ioi^ 2 

 v ' 10 



This formula serves to calculate the dynamic depth D corresponding to any given 

 geometric depth z. 



From this formula we draw as a first approximation 



(a') D = 0.982 



or, solving with respect to z, 



(b') Z=l.02D 



That is, in the case of the sea we have the same approximate difference as in the 

 atmosphere between the figures representing the two kinds of depth amounting to 

 about 2 per cent. 



Solving (tf) by the method employed for (a), section 12, we find the equation 



(b) z = D 0.000000 1 168Z? 2 



by which the geometric value of a given dynamic depth is calculated. 



To make the formulae (a) and (b) suitable for tabulation, we use the same arti- 

 fice as above. Introducing (c), section 12, and neglecting small quantities of the 

 second order, we can write the formulae 



(a") D = {0.982 + 0.00000011012 2 } + o.i(g- 9.80)2 



(b") z = { 1. 0204082? - o.oooooou68Z> 2 } ~^ fo # 



The expressions within the brackets depend on one variable only, and their values 

 are given in tables 3 h and 5 H respectively of the Hydrographic Tables. They give 

 the relation between geometric and dynamic depth in the special case that accel- 

 eration of gravity in sea-level has the value 9.80. The last term in each equation 



