By Charles E. Porter

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DEN. (108) The volume element Ô usually associated with 0 requires a somewhat more complicated discussion which we shall only indicate at the moment. The orthogonal matrix 0 has N(N — l )/2 parameters ah a2, . . , < xn(n- u/2. The volume element Ô is expressed in terms of these parameters. Conven C. E. P orter 33 CHARLES E. , so-called invariant integrals. of Hua15 to whose text we refer the interested reader for a detailed treatment, we proceed in a slightly different manner which does not involve the explicit use of the parametrization of 0 (and hence does not exhibit 6 explicitly) but which does provide us with the energy eigenvalue dependence of the volume element.

Jfe=l (70) Note that rk is not complex conjugated here. The Hermitian conjugate of q is defined as the combination of (69) and (70) 3 = = (/o* I ~ 2 fli* r*. (71) k=l In order to incorporate the quaternion structure into our development we (still following Dyson7) examine the properties of a 2n X 2n matrix which is to be expressed as an n X n matrix Q with quaternion elements qij. Now the transpose operation produces 3 (Q) ij = (qol + ^ qufk)a. (72) k=l This result can be expressed as (Q)ij = 22 “ T 2 f f ,4 T 2 , (73) Statistical T h e o r ie s of Sp e c t r a : F l u c t u a t io n s FLUCTUATIONS OF QUANTAL SPECTRA using the properties of the r-matrices.

In the symplectic case, the condition on S would then read SZS = Z. IV. Average Properties of Spectra It is not our purpose in this article to go deeply into the average prop erties of spectra. , it is intimately associated with the number and type (electron or nucleon, for example) of particle making up the system. 10However, since average spectral quantities form the scale for the fluctuation effects, it is essential that they be mentioned. A detailed presentation of each average property along the 26 Statistical T h eo r ies of S p e c t r a : F lu c tu a t io n s FLUCTUATIONS OF QUANTAL SPECTRA lines of the present volume would probably require a book of the same length as the present one for each property separately.