in Theory and Practice. 417 



change in the distance of the grooves apart, but this effect 

 is vanishingly small compared with the effect on the summa- 

 tion, and can thus be neglected. The displacement is thus 

 proportional to 



Case I. — Lines at Variable Distances. 



In this case we can write in general 



y = an + ain 2 + a 2 n 3 + &c. 



As k, ai, a 2 , &c, are small, we have for the displacement, 

 neglecting the products of small quantities, 



V/ 3 t&[/u(an+a 1 n 2 +a 3 n 3 +&c.)— na 2 n 2 ] 



Hence the term a x n 2 can be neutralized by a change of 

 forms expressed by fiai = tea 2 . Thus a grating having such 

 an error will have a different focus according to the angle n, 

 and the change will be + 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^ 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. Fortunately 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 



2 e »*°» [1 + ib(jm x - tca?y + ibfjLa 2 n* + ib [>a 8 + ib^a x - K a?f] n 4 



+ &c] 



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 /jLa 1 — fca 2 = as before. For the summation we can 

 obtain the following formula from the one already given. 

 Thus 



2 n - 1 ^n = ^P^(n-l). 



sinp 

 Hence 



('2i) m \dp v V s\\\p 



