and on some Phenomena connected titer eivith. 203 



the measurements were ever repeated, it would be advisable 

 to use a collimating lens as well as a more accurate grating. 



The problem of determining the illumination at various 

 points behind a grating exposed to a parallel beam of homo- 

 geneous light, could probably be attacked with success by the 

 usual methods of physical optics, if it were assumed that the 

 grating presented uniform intervals alternately transparent 

 and opaque. Actual gratings, however, do not answer to this 

 description, and, indeed, vary greatly in character. I have 

 therefore preferred to follow the comparatively simple method, 

 explained in my book on Sound, §§ 268, 301, which is ade- 

 quate to the determination of the leading features of the phe- 

 nomenon. 



Taking the axis of z normal to the grating, and parallel to 

 the original direction of the light, and the axis of x perpen- 

 dicular to the lines of the grating, we require a general ex- 

 pression for the vibration of given frequency which is periodic 

 with respect to x in the distance d. Denoting the velocity of 

 propagation of ordinary plane waves by «, and writing 

 k=2tt/X, we may take as this expression 



("LtTX \ 



—j- +e- [ \ cos {Kcd—fJbyz) 



+ B x cos I —j- + e{ \ sm^at—faz) 



. /4c7TX \ , . 



+ A 2 cosl —j-+e 2 ) cos(fcat — /jl 2 z) 



+ B 2 cosf — =- + ej ) sm(fcat— yu 2 z) + , 



where 



2 2 4tt 2 2 2 4.4tt 2 2 2 9.4tt 2 - 



/V=«--^2-> /V = * 2 ^2— 9 /V = K 2 ^2- 7 &C. 



The series is to be continued as long as y? is positive, i. e. as 

 long as the period of the component fluctuations parallel to x is 

 greater than X. Features in the wave-form whose period is 

 less than \ cannot be propagated in this way, but are rapidly 

 extinguished. 



The intensity of vibration, measured by the square of the 

 amplitude, is 



[A + A : COS f—y- + € l J COs(f€Z—/jL l z) 

 -r, (llTX A . , 



+ B 1 cos ( — j- + e{\ sm(/^— /^) 



