236 PROCEEDINGS OF THE AMERICAN ACADEMY 



The values at the point (x, y) of the velocity compoueiits due to 

 any motor at the point (a, h) must be equal respectively to the two 

 velocity components at the point {x — a, y — b) due to the same 

 motor at the origin with axis parallel to its old direction. 



If ^ {x, y), ip (x, y) ,a.ve the velocity potential and flow functions 

 due to the motor at the origin, cb (x — a, y — b), ip {x — a, y — Z>), 

 yield velocity components for the case of the same motor at (a, b) 

 which satisfy the condition stated above. 



If f{z) = (p (x, y) + i if/ (x, y), however, 



f(z— a- bi) = cfi(x — a, y — b) + i . xfj {x — a, y — b), 



and, as a particular case, — — and ~ correspond to two 



^ [z— (a + b i}]" z" 



equal and similarly placed motors, one at the point (a, b), the other 

 at the origin. 



Let us now compare the motors the flow functions of which are 

 the real factors of the imaginary parts of 



i and 'i+li 



z" z" 



respectively ; that is, of 



ces K d -, fi CIS 8 . ces k 6 



where z = p (cos $ + i . sinO), a -\- ^i = p (cos 8 -\- i . sin 8), and 

 where (cos 8 -\- i sin 8) and (cos 6 — i . sin 6) are written cis 8 and 

 ces respectively. 



The two flow functions are 



ft . sm 



sin K ^ J 



and 



'(-!) 



p" p" 



and the values at the point (r, 6) of the velocity components respect- 

 ively parallel and perpendicular to the radius vector drawn from the 



origm are 



K . cos K J K . sin K ^ 

 — — and -j-^— 



pK + l pK+l 



in the case of the first motor, and 



