and

respectively. Here, , , , and are defined in Figure 7.2.

In the limit that the product
remains constant, while
, we obtain a so-called *dipole point source*.
According to the sine rule of trigonometry,

(7.38) |

However, , so we obtain

(7.39) |

In fact, , which leads to

(7.40) |

Thus, in the limit and , we get

Hence, according to Equation (7.37),

Equation (7.36) implies that

(7.43) |

Thus, in the limit and , we obtain

where use has been made of Equation (7.41), as well as the fact that . Figure 7.3 shows the stream function of a dipole point source located at the origin.

Incidentally, Equations (7.26), (7.28), (7.33), (7.35), (7.42), and (7.44) imply that the terms in the expansions (7.23) and (7.24) involving the constants , , and correspond to a point source at the origin, uniform flow parallel to the -axis, and a dipole point source at the origin, respectively. Of course, the term involving is constant, and, therefore, gives rise to no flow.