BAROMETRIC PRESSURES ON THE GREAT LAKES 43 



Then, from (58), it is evident, if one thinks of a step-by-step integration of 

 differences of elevation from the nodal point of the strip to the point under 

 consideration, that /- 



TJ f~i 'v ( z.c\\ 



H-C.Z- (59) 



The summation, indicated by S in this formula, of terms each of the form 



- is supposed to be made in portions of length L so short that along each 

 D 3 



portion D (the depth), may be considered constant. The summation is 

 supposed to extend over the whole distance from the nodal point to the point 

 under consideration. 



Note that if the point under consideration is at a gage, H is the disturbance 

 of the elevation of the water surface at the gage which is produced by the 

 wind. 



DERIVATION OF S x . 



In what has preceded, culminating in formula (59), only a single strip, of 

 unit width, parallel to the direction of the wind has been considered. 



It is evident, if one reviews the various steps by which (59) was derived, 

 that it is true for strips of any one width within which there is no transverse 

 variation of depth. The slope of the water surface along a strip is inde- 

 pendent of the width of the strip provided the depth of water is the same for 

 the whole width of the strip. 



Consider an actual lake to be divided up into strips of moderate width, 

 each parallel to the wind direction. Equation (59) applied to each strip 

 independently would show the values of H differing at adjacent points on 

 adjacent strips. Clearly, if such a condition actually existed for a moment 

 the water would begin to flow across the arbitrarily assumed (but non- 

 existent) boundary between strips, from the higher of the two points to the 

 lower. Such cross-currents would tend always to bring adjacent parts of 

 the various strips to one elevation, and thereby tend to modify somewhat 

 the slope along each strip. 



In any actual lake, with its irregular depths and irregular shore line, there 

 will certainly be such cross-currents. A study of the complicated relations 

 involved leads to the conclusion that such cross-currents are sluggish in 

 general except in shallow water near shore, and that assumption No. 4, 

 stated below, is near the truth and leads to errors so small as to be negligible 

 in comparison with other errors of this investigation which are unavoidable. 



ASSUMPTION No. 4. 



It is assumed that the return gravity current has small components at 

 right angles to the wind direction that cross the boundaries between 

 strips in such a manner as to bring adjacent portions of the various strips 

 nearly to the same elevation and that the slope on each strip thereby estab- 

 lished is the same as if the depth everywhere along each strip were the 

 same as the mean depth for all abreast portions of all strips. For example, 



