in Theory and Practice. 405 



In this case, as in the other, the colours of the spectrum 

 are of variable intensity, and some of them may vanish as in 

 the first example, but the distribution of intensity is in other 

 respects quite different. 



Case II. — Multiple Pekiodic Ruling. 



Instead of having only one groove ruled on the plate in 

 this space a, let us now suppose that a series of similar lines 

 are ruled. 



We have, then, to obtain the displacement by the same 

 expression as before, that is 



smw 



bafi 



j e ib(\n+w) dSf 



. baa 

 Bin -j- 



except that the last integral will extend over the whole number 

 of lines ruled within the space a. 



In the spaces a let a number of equal grooves be ruled 

 commencing at the points y = 0, 'y l9 y 2 , y3, &c, and extending 

 to the points w, y x + w, y 2 + w, &c. The surface integral will 

 then be divided into portions from w to y x , from yi + w to y 2 , 

 &c, on the original surface of the plate for which # = 0, and 

 from w to 0, from y\ + iv to y 1} &c, for the grooves. 



The first series of integrals will be 



I e ib wdy = -! - — } { — e iblxw + e ib ^ 1 — e ibfX ^ + w ) + e ib w* — &c. } 





= -Tj- { — e®^ + ( 1 — e ib ^ w ) (eftw + e ib w* + &c.) + e ib * a ] . 



But ^ a = l since bfia=0 for any maximum, and thus the 

 integral becomes 



{l + e ib w + e ib M* + &c.}. 



The second series of integrals will be 



e ib ^ n+ w)ds{ 1 + e ib w + &c.} 



f 



Jo 



