532 HENRY A. ROWLAND 



. cA) 



1 J * sin 8 



2 



I Q + 



A <-#)(> + 



cos 



-^- [(/* + c' A) (a w) w (// c/)] I . 



This expression is not symmetrical with respect to the normal to the 

 grating, unless the groove is symmetrical, in which case c = c' and 



.=. 



In this case, as in the other, the colors of the spectrum are of vari- 

 able 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 PERIODIC 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 



sn 



2 



r 

 I I e ib ^ x + *) ds, 



* 



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 lt y 2 , y z , etc., and extending to the points w, y l -f- w, 

 y z -f- w, etc. The surface integral will then be divided into portions 

 from w to y u from y l -f- w, to y z , etc., on the original surface of the 

 plate for which x = 0, and from w to 0, from y -\- w to y u etc., for 

 the grooves. 



The first series of integrals will be 



/e 



dv = 4 ( 6 a >i iW + efoMi fc^(i/i+>) + gib^ etc. 

 tOft 



But, e ib ^ a = 1 since &/* = 2^-JV for any maximum, and thus the inte- 

 gral becomes 



f 



< 1 4- fribnyi 4- gib^y^ -\- etc. 



