164 



SMITHSONIAN MISCELLANEOUS COLLECTIONS 



VOL. 51 



ment, but by writing — w in place of w they apply to a descending 



movement. To integrate equation (1) it is necessary to know the 



density as a function of the pressure. We will assume the relation 



given by equation (18) of § 5, and then from equation (20) we shall 



have 



1 



w — w n 



2g 



wa7 



= — z + " 



1 



P 



m — 1 



W 



Wn 



i\ 



(3) 

 (4) 



Here we have supposed p = p and w = iv when z = o. 

 Differentiating equation (18) of §5 we have 



dp _m — 1 dp 

 p m p 



If we differentiate equation (2) and eliminate dp and dp by the 

 aid of equation (1) we shall find 



dw 

 dz 



gw 



m 



(5) 



m — 1 p 



— w 



We conclude from the last equation that the velocity w can never 

 exceed a certain limit w determined by the equation 



it'. 



\ w-1 



(6) 



Introducing the value of - expressed in terms of w and w and 



noting that — is equal to a T Q we find 

 Po 



tn 

 W 



m 



T 



1 n 



m — 1 w 2 



m- 1 



2m-l 



(7) 



If we introduce this value into equations (4) and (3) we shall 

 determine the maximum value of the height z, that a vertical cur- 

 rent cannot exceed, for a given value of the initial velocity w . We 

 see by equation (4) that the velocity increases while the pressure 

 decreases, and equation (3) shows that the pressure p diminishes 

 for increasing values of w. In a vertical ascending current the pres- 



