416 Mr. R. T. Glazebrook on 



P a corresponding point on some other line. Then waves 

 from Q diffracted at A and P respectively will reach Q x in the 

 same phase if QP + Q 1 P = QA + Q 1 A + wA, X being the wave- 

 length. That is, if 



aa>(sin$ — sin^) 



aco 2 ( . , . /cos 2 6 , cos 2 & \ 1 



act) 3 f . , /l a , a 2 * ,\ 

 - -§- | sin <^ - -cos 0+ ^cos 2 0) 



-sin^-^cost+^cos 2 f J j = +»X. . (4) 



This is equivalent to Mr. Baily's formula carried to the next 

 degree of approximation; and his results are obtained by neg- 

 lecting the term in o> 3 and taking <f> and >/r to satisfy the 

 equation „ x 



sin<f>— smy= H (5) 



and then w and v! to satisfy 



JL , r /COS 2 <£ COS 2 ^\ A /Wx 



COS + COS ^Jr— af r H 7-iJrsO. . . (6) 



To consider the aberration we have two cases before us. 

 Let us suppose (1) that equation (5) holds, and determine the 

 value u\j say of v! , considering the terms in o> 3 in equation (4). 

 This will give us what we may call the longitudinal aberration. 



In the second case we shall suppose equation (6) to hold, 

 and determine the value for ty which satisfies (4) to the same 

 approximation. This will give us the lateral aberration. 



In the general case equation (4), as it stands, really deter- 

 mines the locus of the image of Q formed by diffraction at the 

 two lines A and P; and this locus is clearly an hyperbola, 

 with A and P as foci. Waves diffracted at A and P respec- 

 tively will arrive in the same phase at any point of this hyper- 

 bola. For every point such as P on the grating an hyperbola 

 possessing similar properties can be drawn. If all these hy- 

 perbolas meet in a point, then that point is really a focus for 

 waves diverging from Q; they all are in the same phase when 

 they meet there. This is the case if the grating be plane and 

 Q and Q x infinitely distant. If, however, the hyperbolas do 

 not all meet in a point, there is really no focus in its strict 

 sense, only a geometrical focus. If we neglect to 3 and higher 

 terms, then the point given by (5) and (6) is to this approxi- 

 mation common to all the hyperbolas : it is the geometrical 

 focus. 



