OP ARTS AND SCIENCES. 233 



and if Aa be made to approach zero, and fj. to increase so that fi . Aa 

 always has the constant value A, this function approaches as its limit 



If the axis of a simple quadruplet be a diameter drawn through the 

 centre of the shaded portions of the symbol which represents the 

 motor, the expression just found gives, with proper algebraic sign, 

 the flow function due to a simple quadruplet in a circular disk with 

 axis parallel or perpendicular to the radius drawn through the point 

 at which the motor is situated. 



The flow function, which vanishes at every point on the circum- 

 ference, is the real factor of the imaginary part of the function 



w = — X -{ + — . 



('-^) 



,.4 



2 r^ 



{z-df a* /^~_^V «' _^'r (28) 



\ a J a 



The image of the simple quadruplet in question is made up, therefore, 

 of a simple quadruplet and a doublet existing together at the inverse 

 point, with respect to the circumference, of the point where the 

 original quadruplet is situated. 



13. If the axes of the doublets mentioned in the last section had 

 been perpendicular to the axis of x instead of coincident with it, the 

 flow function due to the four doublets would have been 



1, 

 i 



X — a X — a — Aa 



(x — ay + f (x—a — Aay + y' 



2 



^2 



X — — o 



r^ a _ '' 



">-F) 



+ y'' 



The limit of this, as Aa aproaches zero and n Aa is kept equal to 

 the constant X, is 



