GRATINGS IN THEORY AND PRACTICE 



541 



Places. 



= /j. 



Intensities. 



g t 



ba a 



l4 = /JL ^L [J, (&z 1( J ^ (fo 2 ,O] 2 j- Ghosts of 2d order. 



, 5 = ft |* [ J 



'fi /* ^t 7 I 



f Ghosts of 3d order. 



&* 



A* 9 = ,U T-^ 2 



etc. 



etc. 



Each term in this table of ghosts simply expresses the fact that each 

 periodic error produces the same ghosts in the same place as if it were 

 the only error, while others are added which are the ghosts of ghosts. 

 The intensities, however, are modified in the presence of these others. 



Writing ^ = ba lP . and c 2 = ba^. 



The total light, is 









7 



/ 



+ etc. 



which we can prove to be equal to 1. 



Hence the sum of all the light is still unity, a general proposition 

 which applies to any number of errors. 



The positions of the lines when there is any number of periodic 

 errors can always be found by calculating first the ghosts due to each 

 error separately; then the ghosts due to these primary ghosts for it as 

 if it were the primary line, and so on ad infinitum. 



In case the ghosts fall on top of each other the expression for the 

 intensity fails. Thus when e 2 = 2e lt e 3 = 3e^ etc., the formula will 

 need modification. The positions are in this case only those due to a 

 single periodic error, but the intensities are very different. 



Places. Intensities. 



P = - 

 ba a 



