To account for the elementary particles a new physical geometry may be needed. A previous suggestion that this geometry may be finite is followed up by determining the representations of the orthogonal (Lorentzlike) group. Because of the existence of a new type of orthogonality-preserving transformation some of the representations are multiple-valued. A change of value is identified with a gauge transformation and electric charge is recognized as a certain number determining the many-valuedness of the representation. This charge number reverses sign under space inversion. The charged pions and sigma particles are correlated with some of the new representations.

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Computer programs are described which calculate the positive charges contributed by the metallic cations to the surrounding oxygen anions in the low albite, NaA1Si3Os, structure. The unknown variables are taken to be the assumed pure Si-O and A1-O distances, and the coordination number of the sodium ion. The charges are distributed in amounts that are inversely proportional to the cation-anion distances. The most satisfactory charge distribution occurs when Si-O = 1.602( +_ 0.002), A1-O = 1.777( _+ 0-003) A and C.N. = 6 for Na (single atom); the total charges on the four tetrahedral groups of oxygen ions are then: 8.007, 7-999, 8.001 and 7.995 e.s.u. (~r=0"012 e.s.u.) giving a total 'charge unbalance' of 0-014 e.s.u. (o'= 0"024 e.s.u.) and a total A1 content of 1"000 atoms (a= 0"005). It is concluded that the crystal is ideally ionic and that the structure is 'largely ordered' with 82% A1 in one site. It is thought that the small integral C.N. for the Na ion may have important implications for the nature of ionic bonding. An Appendix gives the derivation of relationships for determining the proportion of an oxygen atom shielded by closer atoms from a Na ion.

The connection between Infeld factorization operators and angular momentum operators, well known for spherical harmonics, is extended to other factorization problems by explicitly recognizing them as angular momentum problems. These other problems are: the symmetric top, electron-magnetic pole system, Weyl's spherical harmonics with spin, free particle on a hypersphere. The Kepler problem is also included for it may be thrown into the form of a four-dimensional angular momentum problem. The transformation to momentum space for this problem is very much simplified by the connection between Infeld factorization and angular momentum.

Meson field theory is introduced from an elementary point of view in analogy with familiar classical fields. Three formulations are considered: scalar, pseudoscalar, and vector. In the symmetric theory the isotopic spin of a nucleon is taken to be a classical vector in charge space.