226 PROCEEDINGS OF THE AMERICAN ACADEMY 



AK KA' , . 



y ^ I ^ d I 



1/^0 = /^. Limit J ' JJi A A' 



AA' = 



{x — a)2 + r — y • ^-^' — -4^ (a; — a) 



BL LB' . 



a' 



J3B> = 



{x -OBy + f-y.LB'- BL {x -OB) 



\ X . ., y.cos8+(x — j sinS) 

 J y . cos 8 — (x — a) sin 8 r^ \ a I ' / 1 )\ 



It follows from (1 2) that the image of a doublet of strength /a so situ- 

 ated at a point A — which is at a distance a from the centre, 0, of a 

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

 tended plate — that its axis makes the angle 8 with the radius drawn 



through ^, is a doublet of strength - — and axis making an angle of 



180° — 8 with the line OA, at the inverse point of A with respect to 

 the circumference. 



8. It was long ago noticed * that the functions 



log«, -' 



112 6 



^5' ~3 ' "i' • 



each of which is the derivative with respect to z of the one wliicli 

 precedes it, yield a series of pairs of conjugate functions which repie- 

 sent in order the velocity potential functions and the flow functions 

 due to a source at the origin, to a doublet at the origin, to a quadru- 

 plet t at the origin, to an octuplet at the origin, and so on. 



I will use the word motor to denote in general a source, a sink, 

 a doublet, a quadruplet, or any other combination of sources or sinks 

 at a single point. The strengths of two motors of the same kind 

 shall be in the ratio of /> to q if, when they are similarly placed at 

 the same point successively, they give rise to velocity potential func- 

 tions and flow functions which have values in the ratio oi p to q at 



* See, for instance, Klein's Lectures on tlie Potential, or his paper, " Ueber 

 lliemann's Tlieorie der Alpebraisclien P^inctionen unrl ilirer Integrale." 



t A quadruplet is formed of two equal and opposite doublets in the same 

 manner that a doublet is formed out of a source and an equal sink. An octu- 

 plet is formed in a similar way of two equal and opposite quadruplets, and so on. 



