220 



PROCEEDINGS OP THE AMERICAN ACADEMY 



along a straight line MN such that PM bisects the angle between 

 MN a,\\di the axis of the doublet. In all that follows, the motion will 

 be assumed to be uniplanar. 



3. If, in the subjoined figure, OL ■ ON = OL' • ON' = 0^, aud 

 if there are sources, of strength m, at L' and N', and equal sinks at 



Fig. 3. 



L and iV^, there will be no flow across the circumference drawn around 

 with 0^ as radius. The value at P, the co-ordinates of which are 

 X and y, of the flow function due to these sources and sinks is, if 

 OL = a, LL' = 8, 0A = r, 



i/r „ = m ; tan~^ ^ 



tan 



-1 y 



^y 



= mJtan ^ 



( {x—af—h{x—a)-fif 



X — a 



■tan" 



+ tan 



-1 



y . -tan-^ 



y 



X 



a + S 



X — - 



r^i 



a 



{ax—i^y-\-8x(ax—r')-\-a{a-\-?))y 



'>)A' 



If now 8 be made to approach zero as a limit, and m to increase in 

 such a way that m • 8 is always equal to a constant, /i, the circumfer- 

 ence just mentioned continues to be a stream line and the flow function 

 due to the combination of a doublet of strength /x at X with axis 

 coincident with the radius drawn through Z, and the image of this 

 doublet in the circumference is 



Limit if/p= fx) ^ 



y 



h^)' 



(1) 



+ y 



It is evident from this result (see section 2) that the image of a 

 doublet of strength /x so situated at a point L — wliich is at a distance a 

 from the centre of a circumference drawn with radius r on a tliin plane 

 indefinitely extended plate — that its axis coincides with the radius 



