E.g., C. M. Sparrow, Astrophys. J. 49, 65 (1919).

E.g., H. A. Rowland, Physical Papers (Johns Hopkins University Press, Baltimore, 1902), p. 525.

E.g., P. Frank and R. V. Mises, Differentialgleichungen der Physik (Friedrich Vieweg and Sohn, Braunschweig, 1935), Vol. 2, p. 853 ff.

Lord Rayleigh, Proc. Roy. Soc. (A) 79, 399 (1907).

W. Voigt, Göttinger Nachrichten 40 (1911).

U. Fano, Ann. d. Physik 32, 393 (1938).

U. Fano, J. Opt. Soc. Am. 31, 213 (1941).

A detailed analytical treatment by this method has also been given by C. T. Tai, for the case in which the grating spacing is much shorter than the radiation wave-length, see Tech. Rep. No. 28, Cruft Laboratory, Harvard University, Cambridge, Massachusetts, Jan. 15, 1948.

This set includes waves reflected and transmitted in "complex directions," i.e., waves confined to the proximity of the gratinig's surface. See reference 7, p. 214. For a derivation of the composition of the diffracted light, see reference 6, p. 402.

1O. See reference 7, Appendix.

See reference 1, p. 83. See also reference 14.

An index applied to *v* or *w* means that these functions have the values corresponding to the value of *u* with the same index.

In fact, if no assumption were made at this point, the calculation could be carried further, but more laboriously, taking into account interaction terms between the light diffracted from the grating and that coming from the rest of the metal surface. In that case *r*_{q} and *t*_{q}, should still be considered as slowly varying functions of *u*.

No specific reference is given by Sparrow (see references 1 and 11) for this type of result. A paper by Lord Rayleigh (Phil. Mag. 37, 498 (1919) and *Coll; Works*, Vol. 6, p. 627) is pertinent to this question. A proof of (12) is outlined here. The coefficient *a*_{p} can be written in the form [Equation] If the ruling is perfect except that each groove may be displaced from its theoretical position by a small fraction δ of the spacing L/N and if *p* is a multiple of *N*, *b*_{pm}=(*a*_{p})_{perf} exp(-2π*ip*ô_{m}/*N*), where (*a*_{p})_{perf} is the value of *a*_{p} in the absence of groove displacements. Assuming, finally, that all displacements δ_{m} are distributed as random errors with a r.m.s. value σ, the expected, or mean, value of *a*_{p} is the product of (a_{p})_{perf} and of the expected value of exp(-2π*ip*δ_{m}/N), namely, ∫^{∞}_{-∞}exp[-2π*ip*δ_{m}/*N*-δm^{2}/2σ^{2}]*d*δ_{m}/(2π)½σ=exp(-2π^{2}σ^{2}*p*^{2}/*N*^{2}).