132
BEMARKS CONCERNING FOURIER'S THEOREM AS
ill
in which the first factor expresses total reflexion supposed to originate ic == 0, <£ (as) dec expresses the actual reflecting power at as, and the last fac gives the alteration of phase incurred in traversing the distance 2#. T aggregate reflexion follows on integration with respect to x; with omissi of the first factor it may be taken to be
C + iS,....................................(3]
r +00 |"+°o
where (7= <f>(v)cc>suvdv, S=l (j}(v)sinuvdv, ............(4)
J —CO J —00
with w=47r/X. When c/> is given, the reflexion is thus determined by ( It is, of course, a function of X or u.
In the converse problem we regard (3)—the reflexion—as given for values of u and we seek thence to determine the form of <j> as a functi of as. By Fourier's theorem we have at once
1 I"*3
du {Ccos ux + S sin wr}'.
frH
It will be seen that we require to know G and 8 separately. A knowled of the intensity merely, viz. G2 + $2, does not suffice.
Although the general theory, above sketched, is simple enough, questic arise as soon as we try to introduce the approximations necessary in practi For example, in the optical application we could find by observation t values of G and S for a finite range only of u, limited indeed in eye obs> vations to less than an octave. If we limit the integration in (5) to cor: spond with actual knowledge of C and S, the integral may not go far towai determining (£. It may happen, however, that we have some independe knowledge of the form of <£. For example, we may know that the medii is composed of strata each uniform in itself, so that within each <£ vanish Further, we may know that there are only two kinds of strata, occurri alternately. The value of J^dan at each transition is then numerically t same but affected with signs alternately opposite. This is the case chlorate of potash crystals in which occur repeated twinnings*. Informati of this kind may supplement the deficiency of (5) taken by itself. If it for high values only of u that 0 and S are not known, the curve for <f> fi obtained may be subjected to any alteration which leaves fydx, taken o^ any small range, undisturbed, a consideration which assists materially wh< <f> is known to be discontinuous.
If observation indicates a large G or $ for any particular value of u, infer of course from (5) a correspondingly important periodic term in If the large value of C on: S is limited to a very small range of u, t periodicity of <f> extends to a large range of as; otherwise the interference
* Phil. Mag. Vol. xxvi. p. 256 (1888); Scientific Papers, Vol. in. p. 204.were those which had been contiguous before fracture. That there should be a well-marked difference in this respect between parts both inside a rather small piece of glass is very surprising. I have not again met with this anomaly; but