50 



Mr. R. A. Sampson. 



[Dec. 11, 



whore the integration extends over the area embraced by the 

 circuit. 



This result enables us to transform D^r readily from cylindrical to 

 other systems of coordinates. For instance, consider polar coordi- 

 nates, r, 0, and let us find the circulation in a small rectangle bounded 

 by r, r + dr, 0, + dO. 



Let the velocities in the direction of r and perpendicular to it 

 be R, 9. Then the circulation in this circuit is 



Rdr + Per + -^ (9r) dr~\ d0- |~R +^ di\ dr-Or dO 



. r 



= rdrd0 



\_ 



dQ , e dR~\ 



- --- -=- - 

 i- rddj 



Now 



G = - 



1 (ty 



r sin ' dr ' 



1 dllr 



r~ sin ' dO ' 

 Thus the expression in square brackets is 



or 



p ^ sing d ( l_d+\ 

 dr*^ r 2 dO \sin0d0J 



if p. stands for cos 6. 



Other applications will be found later. 



(6), 



