86 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV. 



in (34) is referred to an origin which is in motion with the velocity 

 V parallel to x. Hence 



d$__dV a? Q__y^4 

 dt dt r dx ) 



where 



dd&amp;gt; dd&amp;gt; n deb . ~ Va* 

 -j = ~ cos 6 j-, sin v 5- cos 20. 

 ax ar rdu r 



^4 



Also 



The pressure at any point of the cylindrical surface is there 

 fore 



COB + F 2 cos 2(9 - I F 2 ......... (36). 



at / 



The resultant pressure on a length of the cylinder is evidently 

 parallel to x\ to find its amount per unit length we must multiply 

 (36) by add . cos and integrate with respect to 6 between the 

 limits and 2?r. The only term which gives a result different 

 from zero is the second, which gives 



-M ............................. (37), 



if M r be the mass in unit length of the fluid displaced by the 

 cylinder. Compare Art. 105. 



If in the above example we impress on the fluid and the 

 cylinder a velocity V in the direction of x, we have the case of 

 a current flowing with velocity V past a fixed cylindrical obstacle. 

 Adding to &amp;lt;/&amp;gt; and ^r the terms Vx and Vy, respectively, 

 we get 



A = _ y( r + L\ cos Q ^ = -v(r- -} sin 0. 

 V r) A r ) 



If no external forces act, and if V be constant, we find for the 

 resultant pressure on the cylinder the value zero. 



87. To render the formula (31) capable of representing any 

 case of irrotational motion in the space between two concentric 

 circles, we must add to the right-hand side the term 



A\nxz ............................ (38). 



