230 PROCEEDINGS OF THE AMERICAN ACADEMY 



and the limit of this as 8 approaches zero is 



X ^ (a:-2 - y-^) sin g - 2 xy cos a — y (x- + f)/' (0)^ . ,^^^. 



{x' + fr ' ^ 



and this is equivalent to a simple quadruplet superposed upon a 

 doublet of strength — A ./' (0) at the origin. 



In general, let fx . vf/ (x, y) be the flow function due to a fixed 

 motor of strength \x.^ and of any kind ; and let this motor be approached 

 (keeping /x 8 constantly equal to A) by an opposite motor of strength 

 \x . /'(8) from a direction making an angle a with the axis of x. The 

 value at the point (x, y) of the flow function due to this second motor 

 i«* the same as the value at the point (.r — o . cos a, 5/ — S . sin a) of 

 the flow function due to the first motor, so that the flow function due 

 to the two motors existing together is 



K [lA (^' y) — /(S) • "A (^ — ^ cos a.,y — Z sin a)], 

 o 



and the limit approached by this as 8 approaches zero is 



\\P^^\, . cos a + Z)^ ./. . sin a — i/. {x, y) ./(O)]. (21) 



11. Let there be two equal opposite motors of the same kind at 

 the points A and J5 in a plane equally distant from an origin, (9, and 

 let the axis of the motor at A make with the straight line OA the 

 same angle that the axis of the motor at B would make with OB 

 if its flow function were the negative of what it really is. If now 

 OB l)e rotated about so as to approach OA as a limit, and if the 

 product of the strength of one of the motors by its distance from the 

 other be kept equal to a constant, A, we get as a limit a kind of motor 

 which sometimes appears in practical problems. 



Let II . R {r, 6) and /* . (^ (r, 0) be the values at the point P, the 

 co-ordinates of which are r and ^, of the velocity components due to 

 the motor at A^ taken respectively parallel and perpendicular to the 

 radius vector, OP. If the angle A OB (see Fig. 10) be a, the values 

 of the velocity components at P due to the motor at Ji are equal but 

 opposite in sign to values at the point P', the co-ordinates of which 

 are r and — a, of the velocity components due to the motor at A. 

 Hence the motors at A and B together cause at P the velocity 

 components 



(x.R(r,6) -i^.R (r, e - a), 



,x . (r, ^) - M . (•'•, 6 — a). 



