PAPER BY PRO*:. OBERBECK. 155 



as the analytical expression for this quantity, omitting a factor that 

 depends upon the adopted units, and in which dn is an element of the 

 normal to the curve whose equation is jt)/p=constant. 



I put 



i dp j. Iftoy.fjp? 



Furthermore, let the velocity of the wind at a point x y be oj so that 

 g?=zu 2 -\-v 2 and let f be the angle between & and y, in which y must 

 indicate the direction of diminishing pressure. Then we have 



uf+vf 



dx Jy 



C08 S= — , . 



■^(SY^SY (0) 



or oj 





If we multiply the first of equation (5) by u and the second by v and 

 add together, there results, 



or 



7 /)0l> i)GO ,")&? 



y COS £ = fa»+ £_+„* +t £_ 



for which by introducing the notation 



we can write 



vcosf = fc(»+~. (8) 



dt v 



From these equations many consequences can be drawn that lead to 

 specially simple theorems when the velocities of the wind are so small 

 that the term 



u- — \-v - - 

 dm dy 



can be neglected. But the following theorems will also be approxi- 

 mately true even if the velocities are larger. 



(a) If we compare an invariable system of wind and one that is va- 

 riable as to its intensity, and of which we will assume that at any given 

 instant there prevails throughout it everywhere uniform velocities and 



