OF ARTS AND SCIENCES. 



231 



Fig. 10. 



The limits of these expressions as a is made to approach zero, while 

 fx . a . OA = A, are 



" and 



A® 



OA OA 



so that the flow function due to the newly found motor is obtained by 

 changing /x to A in the product of the reciprocal of OA and the 

 derivative with respect to 6 of the flow function due to the original 

 motor at A alone ; that is, in 



where 



OA 



fx li = '- and fi ® =: — JJr if/. 



(22) 



A single doublet on the axis of x, at a distance a from the origin, 

 with its axis perpendicular to the axis of x, yields, when combined 

 with another similar motor in the way described in this section, the 

 flow function 



(23) 



^^Xy(x' + f-a') 



a [{X - ay + ff 



This is equal to zero on the axis of x and on the circumference of 

 the circle drawn with the origin as centre so as to pass through the 

 motor. It gives, therefore, a case of flow in a semicircular disk. 



The flow function just found is the real factor of the imaginary 

 part of the function 



A« __^.(1, « ) 



w = — 



(24) 



a(z — oif a i z — a ' (z — af \ 



The motor which corresponds to this flow function may be looked 

 upon, therefore, as formed by superposing a doublet upon a quad- 

 ruplet. 



The image of such a motor (3f) as this in a circumference drawn 

 in its own plane with as centre is a motor of the same kind as M, 



