228 PROCEEDINGS OF THE AMERICAN ACADEMY 



9. The equation "- — = c represents a family of closed curves 



each of which is made up of 2 n equal mutually distinct loops sym- 

 metrically situated about the origin ; each curve passes n times through 

 the origin and each loop is tangent there to two straight lines which 



include an angle of _ . The equation — = c represents the same 



n r" 



IT 



family of curves turned around the orisfin through an angle — . 



Tlie motor which gives rise to the flow function ~ may be 



r 



denoted by the symbol shown in the upper part of Figure 8, in which 



the black portions show the directions in which fluid flows out from 



the point at which the motor is situated, and the open portions the 



directions from which fluid flows towards this point. The motor 



, • , , , _ , . A sin n ^ , , 



which corresponds to the now lunction — may be represented 



similarly by a circle drawn about the origin, and properly divided 

 into n shaded and n unshaded sectors. 



Let the motor A^ which corresponds to the flow function \pJ^ = , 



r" 



be doubled up with an equal opposite motor which approaches it from 

 a direction making an angle a with the axis of x. The flow function 

 i{/^, due to the resulting motor £, is given, apart from a constant factor, 

 by the equation 



ips = Dx\pj^. cos a + Z>j, i/r^ . sin a 



= A xlf^ . cos {6 -a)- ^^ . sin (6 - a) 



n . sin \{n + 1) ^ — «] 



(18) 



Referred to a new initial line drawn in a direction making an angle 



T with the old, this is 



w . pin (n + 1 ) ^ 



^B = ^UT ^ ; (19) 



and it is evident that the orientation of i?'s axis, but not ^'s charac- 

 ter, depends upon a. 



For instance, the flow function due to a unit doublet at the origin 

 with axis coincident with the axis of a: is 



a:' + / 



