OF AKTfc AND SCIENCES. 227 



every point. The unit motor of any kind and the direction of its 

 axis, if it has any, may be chosen at pleasure. 



Let there be in a plane any distribution, A, of n motors, either all of 

 the same kind or of different kinds, of strengths respectively equal to 

 till, ni^, Wg, m„. 



Let there be also a system, B, of n motors, each respectively equal 

 but opposite to a corresponding motor of A, and so situated in the 

 plane that every motor of A could be superposed upon its equivalent 

 in B by moving A parallel to itself through a distance 8 in a direc- 

 tion making an angle a with the axis of x. 



Let u — f(x, y) und ^" = x (^' V) ^^ ^^e values at the point (x, y) 

 of the velocity components due to the system A^ and let (^ (a-, y) and 

 i/f {x, y) be the corresponding values of the velocity potential function 

 and the flow function. 



It is evident that the velocity components due to B have values at 

 (ic, y) equal but opposite to those at the point (x — 8 cos a, y — 8 sin a) 

 of the corresponding velocity components due to A. We may, there- 

 fore, take for the values at any point (x, y) of the velocity potential 

 and flow functions due to A and B existing together, the expressions 



(^ (x, «/) — <^ (x — S cos a, ^ — 8 sin a) ; (13) 



\p{x, y) — ij/ {x — 8 cos a, y — 8 sin a). (14) 



If now Ji be made to approach A, by decreasing 8 and keeping a 

 constant, and if the strength of each of ^'s motors be made to grow so 

 as to keep the product of itself and 8 constant, the expressions just 

 given approach as limits the velocity potential function and the flow 

 function which A would yield if every one of its motors were doubled 

 up with an equal but opposite motor approaching it from a direction 

 which makes an angle a with the axis of x. 



If mi 8 = yu-i, mgS = |Lt2? ^38 = )U,3, etc., where the ya's are con- 

 stant, the limits of the expressions (13) and (14) are the values 

 obtained by changing every m into its corresponding /x in 



Limit {^^^±±^^\ and Limit (^^^ + ^y^' 

 8 = 0^ ^ / 8=^0^ S 



that is, in cos a . D^. ^ -\- m\ a . Dy (ft, (15) 



and cos a . Z)^. xp -\- sma . Dy if/. (1 6) 



If IV =zf(z) := (f) (x. y) + i . if/ (x, y), the function of z which corre- 

 sponds to the new velocity potential function and flow function may 

 be found by changing every m into its corresponding fj. in 



— (cos a -{- i . sin a) . B^ ic. (17) 



