As we cannot have knots in two dimensions, and as Prof. Klein has proved that they cannot exist in space of four dimensions, it would appear that the investigation of their properties belongs to that class of problems for which the methods of quaternions were specially devised. The equation

where ϕ is a periodic function, of course represents any endless curve whatever. Now the only condition to which variations of this function (looked on as corresponding to deformations of the knot) is subject, is that no two values of ρ shall ever be equal even at a stage of the deformation. Subject to this proviso, ϕ may suffer any changes whatever—retaining of course its periodicity. Some of the simpler results of a study of this novel problem in the theory of equations were given,—among others the complete representation of any knot whatever by three closed plane curves, non-autotomic and (if required) non-intersecting.