234 PROCEEDINGS OF THE AMERICAN ACADEMY 



^^ ( f — (x — ay r* 





([(x-af + ff «4- / _^ 



h^J^^ 



This has the value — — at all points of the circumference, and gives 



a^ 



us the flow function inside a circular disk in which there is a simple 



(jiiadruplet with axis making an angle of 45° with the radius drawn 



tlirough it. 



J t is to be noticed that (29) is the real part of the function 





which appears in equation (28). 



14. The flow function due to a simple quadruplet at the centre of 

 a circular disk of radius r may be found by putting a equal to zero 

 in (27). It is 



^ = -2Axy j , ,! ,, -\\^ 



1 2^ 



and the corresponding w is — + - . 



In general, consider the function 



1 , s" r2« + p2" (cos 2 w ^ + I . sin 2 n ^) 



_n „a n 



^m ^2n _ pu (^gQg nO -\- i . sin n 0) 

 r^" . cos nO -\- p-"' (cos 2 n6 . cos ?i ^ + sin 2 ?2 ^ . sin n 9 



+ 



r2» . p" 

 i {p^" (sin 2n 6 . cos w ^ — cos 2 w ^ . sin w 6) — r^" sin w 6] 



^ . cos nO -\- —^—z smn 6. (61) 



