PEOCEEDINGS OF THE SOCIETY. 



731 



the integration. Since a is measured from the edge to the middle of 

 the slit, it is plain that a is the limiting value of /?, and the required 



integral is— 



i: 



= 



2 c . sin a . cos ft . d (3 . = 2 c . sin 2 a. 



This, therefore, is necessarily positive, and does not change its sign 



whatever value (3 may have. Mr. Conrady has tabulated the function 



12 



— sin /8 : a function -which does not seem to have relation to anything 



in particular, but as it, of course, has a series of negative values, he 

 infers that the resulting light phase will also change in the diffracted 

 beam from a broad slit. We see, however, that Mr. Conrady's own 

 postulates lead to a different conclusion, so we need not further discuss 

 this proposition. 



But to return to the spectra of the first order. Mr. Conrady, having 

 seen the beams which are to form these spectra safely started on their 

 way to the focal plane vibrating in unison with 

 the direct light, assumes that they will be in 

 equal phase in the focal plane itself. Now 

 that depends entirely upon the lengths of the 

 optical paths traversed by the direct beam and 

 the diffracted beam respectively in passing from 

 the grating to the focal plane. Mr. Conrady 

 does not appear to have investigated this part 

 of the problem. Prof. Everett did investigate it, 

 and therein lies the explanation of his having 

 arrived at the opposite conclusion. A short 

 examination of this point will suffice to decide 

 the controversy. 



In the following diagram (fig. 124) let G ... CI 

 be the grating on the stage of the Microscope, 

 M x M 2 the middle points of two contiguous 

 slits, s the distance between Mj and M 2 , L the lens, C the prin- 

 cipal focal point, and A x the middle point of one of the spectra of 

 the first order. I use, as far as possible, the symbols used by Prof. 

 Everett in his paper, and accordingly t = the optical distance from the 



grating to the principal focal point C, and T = A. Therefore _, ex- 

 presses this distance in wave-lengths. To express, in like manner, the 

 optical distance from the point M x to the focal point Ai we may write 



L 4. ttl where a x stands for any quantity, integral or fractional, and 



might, if that were possible, stand for 0, so that the use of this expression 

 does not commit us to any proposition concerning the relative optical 

 paths of the direct and diffracted beams respectively. But if this path 

 from 



M x to A! = T -f «i, 



Mj Mi 

 Fig. 124. 



