191 i.l WITH REFLECTING GRATING. 135 



finally 



(8) („ + X)X = . i±i5li^ Jtl) = ^-^cos^i + 0)l2 



^ ^ COS t COS z 



This must therefore be regarded as the fundamental equation of the 

 phenomenon. Equation (7), however, leads on integration to 



(9) N = et2ini/D-]-N,, 

 where N^^D is the width of the crack. 



If the value of A^ from (9) is put into (8) together with the 

 equivalent of X/D, it appears after reduction that 



(;/ -f iVJX, = ^-(005 / -f cos 6) = 2e cos cos . 



The case of A^ = o, ^ > o would correspond to the equation 



(10) h\ = e[i -\- cos (/ -f- 6)] I cos i = 2€ cos^ / cos i, 



which is onl}- a part of the complete equation (8). For i^O, 

 one active half, kh, is necessarily partly behind the other half, k'h', 

 and therefore not adapted to bring out the phenomenon as explained, 

 unless c = o. 



9. Differential Equations. Displaccinoit per Fringe, de/dn. — 

 To test equation (8) or (10) increments must be compared. The 

 latter gives at once since A" is constant relative to e like /, 6, and A, 



de \ X 



('■) 7° .^ ■ = ITe^^^9 



ail cos / + cos V 2 cos cos 



2 2 



which is the interferometer equation when the fringes pass a given 

 spectrum line, like either D line, wdiich is sharp and stationary in the 

 field. Equations (7) and (11), moreover, give after reduction 



i-e 



( 1 2) dNjdn = tan / tan . 



Table I. contains values of de/dn computed from (11), made under 

 widely different conditions (i^6, i<.6, first and second order). 

 The agreement is as good as the small fringes and the difficulty of 

 getting the grating normal to the micrometer screw in my impro- 



