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Quasi-Spherical harmonics, Y m ℓ (θ, φ) are derived and presented for half-odd-integer values of ℓ and m. The form of the φ factor is identical to that in the case of integer ℓ and m: exp (imφ). However, the domain of these functions in the half-odd-integer case is 0≤φ<4π rather than the domain 0≤φ<2π in the case of integer ℓ and m (the true spherical harmonics). The form of the θ factor, P |m| ℓ (θ) (an associated Legendre function) is (as in the integer case) the factor (sin θ) |m| multiplied by a polynomial in cos θ of degree (ℓ−|m|) (an associated Legendre polynomial). A substantial difference between the spherical (integer ℓ and m) and quasi-spherical (half-odd-integer ℓ and m) Legendre functions is that the latter have an irrational factor of √ sin θ whereas the factor of the truly spherical functions is an integer power of sin θ. The domain of both the true and quasi spherical associated Legendre functions is the same: 0≤θ<π. A table of the Associated Legendre Functions is presented for both integer and half-odd-integer values of ℓ and m, for |m| = 0, 1 2 , 1 . . . 11 2 , and for (ℓ−|m|) = 0, 1, 2, 3, 4, 5. The table displays the similarity between the functions for integer ℓ and m (which are well known) and those for half-odd-integer ℓ and m (which have not been recognized previously).

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The optical spectra of the hexaquochromium (III) ion in the host crystals of C(NH2)3Al(SO4)2·6 H2O and several isomorphs, and in AlCl3·6 H2O, are presented and discussed. The broad-band spectra have been analyzed to determine trigonal field parameters, and these results are confirmed by the analysis of the fine-line spectra. The off-diagonal element of the trigonal field, v′, is shown to be as large in magnitude as v, and therefore to be of considerable importance in the interpretation of polarized spectra. Evidence confirming the dependence of the ground-state splitting D on the parameter v′ is also presented.