Interference loith Reflecting Gratings. 127 



This must, therefore, be regarded as the equation of the 

 phenomenon. Equation (7) however leads on integration to 



N=«tani/D + N , ..... (9) 



where N D is the width of the crack. 



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

 the equivalent of \/D } it appears after reduction that 



(7^ + N )X = e(cos^4- cos 0) = 2ecos — — cos-^— . 



The case of N = 0, e>0 would correspond to the equation 

 rik = e(l+ cos (i + 0))l cos i, . . . (10) 



which is only a part of the complete equation (8). In the 

 case of i>0, one active half is necessarily partly behind the 

 other half, and therefore not adapted to bring out the pheno- 

 menon as explained, unless e = 0. 



9. Differential Equations. Displacement per fringe, dejdn. 

 — To test equations (8) or (10) increments must be compared. 

 The latter gives at once, since N is constant relative to e y 

 like i, 6, and X, 



&_ * x (n \ 



dn cos i+ cos 6 _ i + i — 6'' ' ^ ' 

 2 cos -~— cos-— — 



which is the interferometer equation when the fringes pass a 



given spectrum line, like either D line, which is sharp and 



stationary in the field. Equations (7) and (11), moreover,. 



give after reduction 



. i-6 

 cW/dn= tan etan —— (12) 



Values of dejdn computed from (11) agreed as well with 

 observations made under widely different conditions (i>0, 

 i<6, first and second order), as the small fringes and the 

 difficulty of getting the grating normal to the micrometer 

 screw in my improvised apparatus, admit. If this adjustment 

 is not perfect N changes with e. From equation (12) 

 moreover, 



d^ _dNde_Mode_dN, 



dn ~~ d0 dn ~~ dt) dn~ dn 



since N is constant only relative to e when 6 varies. 



10. Deviation per Fringe, Sfc, d6jdn, dd/de. — These 

 measurements are still more difficult in the absence of special 

 apparatus, since e is not determinable and the counting of 

 fine flickering fringes is unsatisfactory ; but the order of 



