224 PROCEEDINGS OP THE AMERICAN ACADEMY 



which agree with the expressions (4) and (5) obtained by using 

 different co-ordinate axes. 



We infer from (7) that the image of a doublet of strength jx so 

 situated at a point A — which is at a distance a from the centre of a 

 circumference drawn with radius r on a thin plane indefinitely ex- 

 tended plate — that its axis is perpendicular to the radius vector 



o 



drawn through A, is a doublet, of strength fx — and axis parallel to 



that of the first doublet, at the inverse point of A with respect to the 

 circumference. 



If we make a equal to r, (7), (8), and (9) give us the flow function 

 and the velocity components inside a circular plate when there is a 

 doublet at some point of the circumference with its axis coincident with 

 the tangent to the plate at the given point. The forms of these expres- 

 sions might have been inferred from the results given in section 2. 



6. It is to be noticed that if in Figure 6 O'B' and OB are parallel 

 lines, and 00', AA', and £B' perpendicular to them, and if there are 



f-0' Q-A^ B^ 



G-0 ^A B-# 



Fig. 6. 



sources of strength m at 0', A, and B, and sinks of equal strength m 

 at 0, A', and B\ a circumference drawn around as centre with 

 radius ^/OA . OB will form part of a line of flow due to the combina- 

 tion of the sink at and the sources at A and B, but not one of the 

 lines of flow due to the combination of three doublets obtained by 

 making O'B' approach OB and increasing m at the same time so that 

 m . 00' shall always be equal to the constant /x. 



7. In Figure 7 let OA.OB= OA' . OB' =0C^= r\ and OA = a, 

 and let the curve AA' make with OM the angle 8. If, then, there are 

 sources of strength m at A' and B', and sinks of strength ;» at A and B, 

 one of the lines of flow will consist in part of the circumference (C) 

 drawn with radius r about as centre. If if/p is the value at P of the 

 flow function due to this combination, the limit appronchod by ij/p as 

 A' is m ide to approach A along the curve A'A, and Ji' to approach 

 ^ along the corresponding reciprocal curve B' B, will be the value at 

 P of the flow function due to a doublet at A, the axis of which makes 

 the angle 8 with the radius drawn through A of the circumference C, 

 and to the image of the doublet in C. 



