OP ARTS AND SCIENCES.' 229 



If this motor be doubled up with its negative approachins; from tlie 

 positive purt of tlie axis of x, the resultaut quadruplet (see Fiy. 8) 

 corresponds to the flow function 



(x- + y^Y 



e « 



Fig. 8. Fig. 9. 



If, on the other hand, it be doubled up with an equal opposite 

 doublet (see Fig. 9) approaching it from the positive part of the axis 

 of y, the resultant quadruplet corresponds to the flow function 



•2\2 



One of these quadruplets is evidently equivalent to the other turned 

 through an angle of 45°. 



10. Besides the simple quadruplets obtained by combining two 

 equal and opposite doublets just as a doublet is formed from a source 

 and a sink, other combinations of two doublets sometimes appear 

 when one attempts to find the image of a simple motor in the cir 

 cumference of a circle in its plane. 



In one common case, a fixed doublet D^ of strength fx. is ap^ 

 preached by another doublet D^, the axis of which remains always 

 parallel but opposite in direction to that of D^. The path of D.^ is a 

 straight line, but its strength is equal to fxf{S), where B is the dis- 

 tance between the two doublets at any time, and f (S) is a finite, 

 continuous, and single-valued function such that ^(0) =: 1. Th^ 

 product of fx and 8 is kept always equal to a constant, A. 



The flow function due to a doublet of strength fx at the origin 



with axis coincident with the axis of x is -J~^— , and the flow 



x^-\-y^^ 



function due to a reversed doublet of strength (xf (8) at the point 

 {x — 8 . cos a, y — 8 . sin a) 



— fx . (y — 8 sin a) ../'(8) 



IS 



(a; — o . cos ay -\- {y — 8 . sin a)^ 



The flow function due to the two doublets existing together is 



A ( y _ (;/ — 8 . sin a) /*(8) ) 



8 Ix^ -\- y'^ (x — 8 . cos a)'^ + (y — & . sin a)' \ 



