The actual form of the set of linear substitutions on 3 symbols, which gives a representation of the simple group of order 168, has been known for many years.

In these Proceedings (vol. 20, pp. 247–9) I have determined the form of the set of linear substitutions on 5 symbols which gives a representation of the simple group of order 660.

If G is a group of finite order which contains an operation P of prime order p, permutable only with its own powers, the order of G must, by Sylow's theorem, be of the form (1 + kp) pś, where s is a factor of p – 1. The greatest subgroup of G, which contains self-conjugately {P};, the subgroup generated by P, must be a metacyclical subgroup {S, P}, where

while g is a primitive root of the congruence gs = 1 (mod. p).

§ 1. Bohr's original (1913) theory of the stopping power of light atoms for α-particles seems to invite further study. It is well-known that in this theory, which exhibits a general agreement with experiment, the loss of energy by the α-particle is calculated on purely classical principles, as the energy transferred to electrons initially at rest; and elastically bound to atoms. A large part of the loss so calculated (about one-half) comes from transfers to electrons in distant atoms of amounts less than the energy required to transfer those atoms to their next higher stationary state. This has created a serious difficulty according to more recent views of the quantum theory in which it has been presumed that any effective interaction must always leave the atom in a stationary state, and has given rise to the investigations of Henderson, which however do-not agree with experiment. In some recent speculations Bohr has suggested a possible way of escape from this difficulty, of great theoretical interest, to which he has been led by somewhat similar difficulties in the theory of radiation. He suggests that there may be reason to expect that energy may be only statistically conserved in just such a type of interaction by the atomic switches, while the classical calculation for the α-particle ought to retain its validity.