An attempt is made to simplify the theory of the magnetic field round a plane circular current filament. The lines of force round such a filament are identical with the lines of flow round a circular vortex in hydrodynamics. A diagram showing these curves was drawn by Donald Macfarlane under Kelvin's direction in 1869 and has often been reproduced since. The author shows how the curves can be drawn by means of simple bipolar formulae. He shows that the mutual inductanceMbetween two concentric and coplanar circles the radii of which are a and b is given very approximately by the formulaM = 8Π2b2/a + 3√(a2-b2)Ifb/ais less than 0.5 the maximum inaccuracy of this formula is less than 1 in 40 000. Ifb/ais less than 0.7 its inaccuracy is less than 4 in 1 000. It is less than 1 per cent whenb/a= 0.9. Even whenb/ais 0.95 the inaccuracy is less than 2 per cent. A formula is given for the attraction between two coaxial circular filaments. An equation is also given from which the distance between them when their attraction is a maximum can be computed. If 6 is not greater than about the tenth part ofa, the distanceybetween them, when their attraction is a maximum, is given very approximately by2y = a + b.In a mathematical Appendix a fairly complete list is given of formulae for the two fundamental elliptic integrals and of formulae for differentiating them. Some of these formulae are new and some can only be found in treatises which are out of print and are obtainable in very few libraries. They can be usefully applied in electrical theory