[phi] + [psi]i = [f](x + yi), [Nabla]²[psi] = 0, [Nabla]²[phi] = 0; (6) or putting [phi] + [psi]i = w, x + yi = z, w = [f](z). The curves [phi] = constant and [psi] = constant form an orthogonal system; and the interchange of [phi] and [psi] will give a new state of uniplanar motion, in which the velocity at every point is turned through a right angle without alteration of magnitude. For instance, in a uniplanar flow, radially inward towards O, the flow across any circle of radius r being the same and denoted by 2[pi]m, the velocity must be m/r, and [phi] = m log r, [psi] = m[theta], [phi] + [psi]i = m log re^(i[theta]), w = m log z. (7) Interchanging these values [psi] = m log r, [phi] = m[theta], [psi] + [phi]i = m log re^(i[theta]) (8) gives a state of vortex motion, circulating round Oz, called a straight or columnar vortex. A single vortex will remain at rest, and cause a velocity at any point inversely as the distance from the axis and perpendicular to its direction; analogous to the magnetic field of a straight electric current. If other vortices are present, any one may be supposed to move with the velocity due to the others, the resultant stream-function being [psi] = [Sigma]m log r = log [Pi]r^m; (9) the path of a vortex is obtained by equating the value of [psi] at the vortex to a constant, omitting the r^m of the vortex itself. When the liquid is bounded by a cylindrical surface, the motion of a vortex inside may be determined as due to a series of vortex-images, so arranged as to make the flow zero across the boundary. For a plane boundary the image is the optical reflection of the vortex. For example, a pair of equal opposite vortices, moving on a line parallel to a plane boundary, will have a corresponding pair of images, forming a rectangle of vortices, and the path of a vortex will be the Cotes' spiral r sin 2[theta] = 2a, or x^(-2) + y^(-2) = a^(-2); (10) this is therefore the path of a single vortex in a right-angled corner; and generally, if the angle of the corner is [pi]/n, the path is the Cotes' spiral r sin n[theta] = na. (11) A single vortex in a circular cylinder of radius a at a distance c from the centre will move with the velocity due to an equal opposite image at a distance a²/c, and so describe a circle with velocity mc/(a² - c²) in the periodic time 2[pi](a² - c²)/m. (12) Conjugate functions can be employed also for the motion of liquid in a thin sheet between two concentric spherical surfaces; the components of velocity along the meridian and parallel in colatitude [theta] and longitude [lambda] can be written d[phi] 1 d[psi] 1 d[psi] d[psi] -------- = ----------- ---------, ----------- --------- = - --------, (13) d[theta] sin [theta] d[lambda] sin [theta] d[lambda] d[theta] and then