GRATINGS IN THEORY AND PRACTICE 543 



Hence the term a^n 2 can be neutralized by a change of forms ex- 

 pressed by fjia 1 =K a 2 . Thus a grating having such an error will have 

 a different focus according to the angle n, and the change will be -f- on 

 one side and on the other. 



This error often appears in gratings and, in fact, few are without it. 



A similar error is produced by the plate being concave, but it can 

 be distinguished from the above error by its having the focus at the 

 same angle on the two sides the same instead of different. 



According to this error, a^n 2 , the spaces between the lines from one 

 side to the other of the grating, increase uniformly in the same manner 

 as the lines in the B group of the solar spectrum are distributed. For- 

 tunately it is the easiest error to make in ruling, and produces the least 

 damage. 



The expression to be summed can be put in the form 



ib (>! a 2 ) n 2 + ib/tatf + ib Oa 3 -f ib (/^ <*)'] n*+ etc.] 



The summation of the different terms can be obtained as shown 

 below, but, in general, the best result is usually sought by changing 

 the focus. This amounts to the same as varying K until //a x *a 2 = 

 as before. For the summation we can obtain the following formula from 

 the one already given. Thus 



Hence 



vn -\ e Zipn sm n P e ip (n - 1). 



sin p 



= e p( -i) 



m 



dp J I amp 



When n is very large, writing *^ = pn = n Nn -f q, we have 



dq 

 Whence writing 



c = 



c' =. /JLCii, 



c" = b &a 3 + i 



c'" = etc., 



