Fourier 9 s Theorem as applied to Physical Prohlems. 865 

 where 



/* + « /* + oo 



0=1 <f)(v) cos uvdv, S— 1 <£(v) sin iwdv, . (4) 



»/ — 00 «■ 00 



with u = 4z7r/\. When cf> is given, the reflexion is thus 

 determined by (3). It is, o£ course, a function of X, or n. 



In the converse problem we regard (3) —the reflexion — as 

 given for all values of u and we seek thence to determine 

 the form of <f> as a function of oc. By Fourier's theorem we 

 have at once 



«„)= ij; 



du{C cos w,i?+ S sin«#}. . . (5) 



It will be seen that we require to know C and S separately. 

 A knowledge of the intensity merely, viz. C 2 + S 2 , does not 

 suffice. 



Although the general theory, above sketched, is simple 

 enough, questions arise as soon as we try to introduce the 

 approximations necessary in practice. For example, in the 

 optical application we could find by observation the values 

 of G and S for a finite range only of u, limited indeed in eye 

 observations to less than an octave. If we limit the inte- 

 gration in (5) to correspond with actual knowledge of C and 

 S, the integral may net go far towards determining <fi. It 

 may happen, however, that we have some independent 

 knowledge of the form of cf>. For example, we may know 

 that the medium is composed of strata each uniform in itself, 

 so that within each (j> vanishes. Further, we may know that 

 there are only two kinds of strata, occurring alternately. 

 The value of ^<f>dx at each transition is then numerically the 

 same but affected with signs alternately opposite. This is 

 the case of chlorate of potash crystals in which occur repeated 

 twinnings *. Information of this kind may supplement the 

 deficiency of (5) taken by itself. If it be for high values 

 only of u that C and S are not known, the curve for <p first 

 obtained may be subjected to any alteration which leaves 

 ^<f>dx, taken over any small range, undisturbed, a considera- 

 tion which assists materially where cf> is known to be dis- 

 continuous. 



If observation indicates a large C or S for any particular 

 value of u, we infer of course from (5) a correspondingly 

 important periodic term in $. If the large value of C or S 

 is limited to a very small range of u, the periodicitv of <f> 



* Phil. Mag. vol. xxTi. p. 256 (1888) ; Scientific Papers, vol. lii. 

 v. 204. 



