44 PROCEEDINGS OF THE AMERICAN ACADEMY. 



This is not the case, as may be shown by trial. If there are to be but 

 four reproductions and if the relative dispersions and intensities of these 

 reproductions are to be explained, the values 70 and 3 give the result 

 best in agreement with experiment. 



Up to this point it has been the object of the discussion to contrast, in 

 the light of these experiments, the theory of Professor Runge with that 

 part of the theory as given by Rowland which would seem most obviously 

 applicable to the case. This part of the theory was called '' One line in 

 m displaced.* " The other part of the analysis which deals with " ghosts " 

 came under the title " Periodic Error."t A moment's consideration will 

 show that the lines under discussion are not of this latter class. 



In the case of Simple Periodic Error the position of any groove in the 

 grating ruling is given by the equation, 3/ = a^n -f a^ sin (en). Thus 

 the nth groove from a fixed line of reference is out of its true position by 

 a term which varies as a sine function with period e ; the maximum value 

 of the displacement is of course a^. Thus no groove in the grating 

 surface is exactly in its proper place unless sin (e n) = 0. The system 

 of ghosts to which this form of error gives rise is characterized by the 

 following properties.^ Ghosts of any order must occur in pairs. Of a 

 pair one lies to the right of the parent line, the other to the left. Ghosts 

 of the second order must lie exactly twice as far from the real line as 

 ghosts of the first order ; ghosts of the third order three times as far. 

 The distance of a ghost of a given order from its parent line is a constant 

 independent of the order of the spectrum in which the parent line is 

 measured. These three properties are not in any way possessed by the 

 lines under discussion. This, together with the fact that no numerical 

 application of the theory of ghosts to the case in hand seems possible, 

 excludes it from further consideration. 



In short, then, Professor Runge's explanation of the false spectra Seems 

 to fit the facts most accurately. It is perfectly possible to extend the 

 theory to even the more complex cases where there are more than four 

 reproductions of every real line. The period of the error and the value 

 of n may be taken at pleasure, so that the treatment can be made to fit a 

 great variety of cases. In practice, however, the errors of ruling in 

 those gratings which give a great number of faint false spectra are too 

 complex to make calculation profitable. 



It may he of interest to remark that false spectra are not confined to 

 concave diffraction gratings, but are to be found in the spectra produced 



* Rowland's Pliysical Papers, p. 535. t Ibid., p. 53G. | Ibid., p. 519. 



