into small squares. The procedure was discussed in detail by Closterhalfen,^^ and a machine 

 for use in such graphical constructions was described by Fb'ttinger.^^ 



Obviously itself could be the velocity potential for another type of irrotational flow, 

 satisfying a different set of boundary conditions. The stream function to accompany it would 

 then be -0. For, if the new potential and stream function are <^' = <//, i// '= - 0, by Equations 

 [13f, g] 



d(f) d'Jj ' d(f) ' dib ' 



dx dy dy dx 



which are simply Equations [13f, g] written for </> ' and i/y'and show that these functions stand 

 in the relation to each other that is characteristic of a potential and its associated stream 

 function. 



Thus the two-dimensional types of irrotational flow occur in associated pairs, which 

 might be called conjugate pairs. At a given point, velocities in two conjugate flows have per- 

 pendicular directions but equal magnitudes; in fact, the vector velocity in the second type is 

 merely rotated, relative to that in the first type, through 90 degrees in the countercloci<wise 

 direction, or from x toward y. For, in the second type of flow the components are 



, dcf) dih d(f> ' d\lj 



u = = ~=-v, v' = = - = u [13j, i<] 



dx dx d y d y 



by Equations [13a, b]; the magnitude of the velocity is thus q = {v? + «^)'''% and the directions 



are as stated, as is illustrated in Figure 12. 



All of the equations written down in this section are linear and homogeneous in the 



dependent variables. For this reason it is easily seen that if 0^, i^j are the potential and 



streaiii function for one type of irrotational flow and (/>„, li- for another, then the sums, 



, = oi J + <i 2 ) '1^ 3 ~ 'I' I + 'Z* 2 



represent the potential and stream function for 

 a third possible type. In tlie latter type, which 

 is said to be formed by superposition of the 

 first two, the velocity as a vector is easily 

 seen to be the vector sum of the two component 

 vector velocities. Again, both potential and 

 stream function may be multiplied by the same 

 constant. 



Finally, if 



^1 

 dx 



dx 



Figure 12 — Relation between particle 

 velocities in two conjugate flows. 



23 



