216 Michelson — Spectroscope without Prisms or Gratings. 



came ; for that the grooves must be correspondingly deep, and 

 the grating space would vary with the depth. Fig. 1 at once 

 suggests a possible method of effecting the same result, by 

 building up the steps by equal thicknesses of optical glass. ' 

 Here the difficulty, even supposing the optical work to be 

 practically perfect, would be the joining of the separate plates 

 in such a way as to have always the same distance between 

 them. 



By using the same arrangement for transmission instead of 

 reflection this difficulty is avoided — and there remains abso- 

 lutely nothing but the difficulty of making a considerable num- 

 ber of plane-parallel plates of the same thickness — to an order 

 of accuracy only one-fourth that required in the former arrange- 

 ment, or even one-tenth of this if the other medium be water 

 or oil instead of air. 



Probably the surprising thing is the smallness of the num- 

 ber of plates required to give results which are comparable 

 with those of the best gratings. This can be shown as follows : 

 2# let abd (fig. 2) be one step 



in the series of plates and let 

 ab — s and bd—t. If m is 

 the order of the spectrum 

 observed, mX = fi . bd—ac or 



mX = pt—tcosO + 8sin0. 



dfx 



dO/dX 



dO i / dm 



m—t 



dX 



£sin0 + Scos0 



£sin# + Scos0 



and if Bd is the displacement 

 corresponding to BX and £0 t 

 that corresponding to 8m=l y . 

 assuming Cauchy's formula 

 [a = a + b/X* . and taking the 

 approximate value of m = 

 (/jL — l)t/X, we have 



t 8X 

 W/M.ssKp-l) +2(,*-a)]-. r 



For ordinary glass the coeflr 



cient of t/X is approximately equal to unity ; so that if — = 



•001 as in the case of the two sodium lines, and t — 5 mm = 

 10000 X, then dO = 10 d0 19 that is, the sodium lines would be 

 separated by ten times the distance between the spectra. 



