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By Jean Perrin

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1) Let vector n may be expressed in the frame (I, J, K): n = cos θˆ K + sin θˆ U, U = cos ϕˆ I + sin ϕˆ J. 3) ⊥ ⊥ e⊥ 1 = cos θ0 I, e2 = 0, e3 = sin θ0 I, u⊥ = cos θ0 cos ϕ I + sin ϕ J, v⊥ = − cos θ0 sin ϕ I + cos ϕ J. 4) that the component of T⊥ 1 (±k) upon I and the (±k) upon J are null for the three cases = m1 − m2 = 0, 1, component of T⊥ 2 or −1. 11) 1. 5) ˆ e±ikr cos θ (a(r, θ) cos ϕ cos θ0 + c(r, θ) sin θ0 )dτ ]I. 6) 2. 7) ˆ e±ikr cos θ (( b(r, θ) sin2 ϕ + a(r, θ) cos2 ϕ) cos θ0 + c(r, θ) cos ϕ sin θ0 )dτ ] I.

3 The Solution of the Quaternionic Equation 25 Since ne23 = n, nue3 = −w, we obtain κ ≤ −1 : m m m m m Nm 1+κ = L1+κ e3 + M1+κ u ⇒ N 1−κ = L1+κ n − M1+κ w. 22) that we can write N −m−1 = v × Nm 1+κ . 3 Expression of the Solutions by Means of the Legendre Polynomials We use the normalized associated Legendre polynomials Plm (x) = (−1)m (l − m)! (l + m)! 1/2 2l + 1 2 1/2 m+l 1 [(x2 − 1)l ] 2 m/2 d [1 − x ] . 2l l! dxm+l They verify the two relations (see [26], p. 28) and the recursion one [(l + m + 1)(l − m)]1/2 Plm+1 (cos θ) + 2m cot θPlm (cos θ) +[(l − m + 1)(l + m)]1/2 Plm−1 (cos θ) = 0 .

254) and the anomal (Sect. 5) Zeemann components of the transitions between two levels of energy E1 , E2 . 30) 0 corresponding to an average over all the directions of the vector k for a given value of k, and for two given states Ψ1 , Ψ2 . It has been established in [8] that, in the Schr¨ odinger theory, for a fixed couple (E1 , E2 ) of levels, the values of S are the same in the three cases m1 − m2 = 0, m1 − m2 = 1, m1 − m2 = −1. This property may be extended [15] to the Dirac theory in the following way.

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