{x,y) 



(^0-%^ 



X Figure 44 - Definition of a, x^, y^, in Equation 



[37r] , 



38. LINE QUADRUPOLE 



^2 



wliere A is a real constant. If 2 = re '^, 



hence 



0+zi/f = 4(re'^ =— (cos 2 ^ - z sin 26) 



A A . y 



= -y cos 2(9, lA = - — sin 2^, 6 = tan"^ — 



dw 2A 



- u + iv = — =-— — 



d^ r3 



2-4 



— (cos SO - i sin 3^) 



2A 



2 A 



cos 30, v =— sin 3( 



[38 a] 



[38b,c] 



[38d] 



[38e,f] 





[38g,h] 



This type of flow can be produced in the limit out of that due to two opposing dipoles 

 placed close together, hence the name quadrupole. The origin is a singular point, a pole of 

 the second order. 



The equipotential curves and streamlines are leniniscates. The first quadrant of the 

 flow net is illustrated in Figure 45; other quadrants are geometrically similar, with changes 

 of sign in and ih that are easily determined. 



( For notation and method; see Section 34; Reference 1, Article 63.) 



78 



