Bertrand proved in 1842 that when the impulses applied are given in amount, and are applied at specified points, the system starts off with kinetic energy greater than that of any other motion which is consistent with the given impulses and the connections of the system. This other motion must be such as could be produced in the system by the given impulses, together with any other set of impulses capable of doing no work on the whole. Thomson's theorem is curiously complementary to Bertrand's. Let the system be acted on by impulses applied at certain specified points, and by no other impulses of any kind; and let the impulses be such as to start those selected points with any prescribed velocities. The system will start off with kinetic energy which is less than that of any other motion which the system could have consistently with the prescribed velocities, and which it could be constrained to take by impulses which do no work on the whole. In each case the difference of energies is the energy of the motion which must be compounded with one motion to give the other which is compared with it. A simple example, such as might be taken of the particular case considered by Euler, may help to make these theorems clear. Imagine a straight uniform rod to lie on a horizontal table, between which and the rod there is no friction. Let the rod be struck a blow at one end in a horizontal direction at right angles to the length of the rod. If no other impulse acts, the end of the rod will move off with a certain definite velocity, and the other parts of the rod (which is supposed perfectly unbending) will be started by the connections of the system. It is obvious that any number of other motions of the rod can be imagined, all of which give the same motion of the extremity struck. But the actual motion taken is one of turning about that point of the rod which is two-thirds of the length from the end struck. If the reader will consider the kinetic energy for any other horizontal turning motion consistent with the same motion of the end, he will find that the kinetic energy is greater than that of the motion just specified. This motion could be produced by applying at the point about which the rod turns the impulse required to keep that point at rest. The impulse so applied would do no work. The actual value is 1⧸8mv², where m denotes the mass of the rod and v the velocity of the end. If the motion taken were one of rotation about a point of the rod at distance x from the end struck, the kinetic energy would be m(4l² － 6lx + 3x²)v²⧸6x², where 2l is the length of the rod, and this has its least value 1⧸8mv² for x = 4l⧸3. For example, x = 2l gives 1⧸6mv², which is greater than the value just found.