ii 4 



DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



This outflow has a simple additive property. Let the considered area be divided 

 by a line into two parts (fig. 93). The transport through the dividing line will then 

 appear in the expression for the outflow out of each part. But in the sum of these two 

 outflows this transport will drop out, as it represents the transport out of the one 

 and into the other area. The sum of the outflows out of the two parts will therefore 

 be equal to the outflow out of the total area. As each part can be divided again, 



and so on, we get the general result that the outflow out 

 of all the parts into which an area can be divided will 

 be equal to the outflow out of the total area. We sym- 

 bolize this result by the equation 



(b) fAds = VJA n ds 



the first member being extended to the contour of the 

 total area, and the integrals in the second member being 

 extended to the contours of all the parts into which the total area has been divided. 

 The division may be continued indefinitely. The areas of which the contours 

 appear in the second member of equation (b) may therefore be considered as elemen- 

 tary areas da. As they can have any form let them be limited by the two elements 

 ds and ds' of two vector-lines, and by the two elements dn and dn' which are normal 

 to these lines (fig. 94) . The outflow will be the difference between the transport^ 'dn' 

 and A dn through the latter elements, 

 (c) A'dri - Adn 



Here A ' will vary as we proceed along a vector-line s, and the same will be the case 

 with the normal distance dn' between the two vector-lines. We may then consider 

 these quantities as functions of 5 and use the developments 



A' = A+ 9 -^ds 

 3s 



dn' = dn-\--~ds 

 2s 



dn\ 



ds 



Fig. 94. Fig. 95- 



When we introduce this and disregard quantities of the second order we get as 

 expression of the outflow 



3 A , , , . 3dn , 

 dsdn+A r-ds 



CO Ci 



or, when we separate the factor da = dnds which represents the area of the element, 

 we get the expression of the outflow in the form 



(2A , . 1 2dn\ , 



