Bertrand's theorem applied to this case of motion is not quite so easy, perhaps, to understand. The motion which is said to have maximum energy is one given by a specified impulse at the end struck, and this, in the absence of any other impulses, would be a motion of minimum energy. But let the alternative motion, which is to be compared with that actually taken, be one constrained by additional impulses such as can together effect no work, and the existence of the maximum is accounted for. The kinetic energy produced is one-half the product of the impulse into the velocity of the point struck, that is ½Iv, and it has just been seen that this is the product of (1⧸6)mv² by the factor (4l² － 6lx + 3x²)⧸x². This factor is 3I⧸mv, and is a minimum when x = 4l⧸3. Thus for a given I, v will have its maximum value when the factor referred to is least, and ½Iv will then be a maximum. The bar can be constrained to turn about another point by a fixed pivot there situated. An impulse will be applied to the rod by the pivot, simultaneously with the blow; and it is obvious that this impulse does no work, since there is no displacement of the point to which it is applied. The two theorems are consequences of one principle. The constraint in each case increases what may be called the effective inertia, which may be taken as I⧸v. Thus when v is given, I is increased by any constraint compelling the rod to rotate about a particular axis, and so ½Iv, or the kinetic energy, is increased. On the other hand, when I is given the same constraint diminishes v, and so ½Iv is diminished. A short paper published in the B. A. Report for 1852 points out that the lines of force near a small magnet, placed with its axis along the lines of force in a uniform magnetic field, as it would rest under the action of the field, are at corresponding points similar to those of the field of an insulated spherical conductor, under the inductive influence of a distant electric change. Further, the fact is noted that, if the magnet be oppositely directed to the field, the lines of force are curved outwards, just as the lines of flow of a uniform stream would be by a spherical obstacle, at the surface of which no eddies were caused. This is one of those instructive analogies between the theory of fluid motion and other theories involving perfectly analogous fundamental ideas, which Thomson was fond of pointing out, and which helped him in his repeated attempts to imagine mechanical representations of physical phenomena of different kinds.