OF ARTS AND SCIENCES. 



219 



as a limit a "doublet"* of strength fj., the axis of which is PX, the 

 limiting position of the straight line drawn from P to P'. We shall 

 find it convenient to represent sources and sinks respectively by black 

 and unshaded circles, and doublets by circles half black and half un- 

 shaded. The black portion of a doublet circle indicates the directions 



in which there is a flow awai/ from the point where the doublet is 

 situated ; the unshaded portion indicates the directions from which 

 there is a flow towards this point. The axis of a doublet bisects both 

 the black and unshaded portions of the doublet circle. (See Fig. 1.) 

 If P be used as origin and PX a.s axis of abscissas, the velocity poten- 

 tial function due to the doublet is <^ = — ^ 



„ „ and the flow func- 

 If X -{- yi ^ z, these are respectively the real 



tion IS i/^ = „ , ., 



part and the real factor of the imaginary part of the function ^ 



z 



Fig. 2. 



The equipotential lines and the lines of flow are circles (see Fig. 2) 

 touching the axes of ^^ and x respectively at the origin. 



In uniplanar motion, the velocity at a point 31 due to a doublet of 



strength ^u, at a point P is numerically equal to ^ ^ , and is directed 



* Basset, Treatise on Hydrodj'namics, Art. 47 ; Mascart and Joubert, Trea- 

 tise on Electricity and Magnetism, Art. 151; Neumann, Untersiichungen iiber 

 das Logarithmische und Newton'sclie Potential Chap. TV.- etc.; etc. 



