386 Transactions of the Society. 



of wave-length X ; and the disturbances due to C, A l} B 1} A 2 , B^ 

 are represented by the expressions 



t 



C COS 2 IT „ 



T 



a Y cos 2 it (^ + a x ) ; a 2 cos 2 7r ( ~ + a 2 ) 5 



lh cos 2 w (^ - &J ; Z> 2 cos 2 ?r (^ - A ) ; 



c, « 1; l>i, a 2 , b 2 being constant coefficients ; and a 1} /3j, a 2 , /3 2 con- 

 stants depending on the position of the grating. 



Now consider the effect of a small displacement x of the 

 grating in its own plane, perpendicular to the bars. As all the 

 points P, 0, A lt B x , A 2 , B 2 are fixed, there will be no changes in 

 the distances of the spectra from P. Let s denote the grating- 

 interval, and let M be the middle point between a selected pair 

 of consecutive bars. If the grating were displaced through the 

 distance s, the optical path through M from the incident wave- 

 front to A j. would be increased or diminished by \, and an equal 

 and opposite change would be made in the path to B x . For A 2 

 and B 2 the changes would be ± 2 \. The change of path for 

 Aj and B x due to displacement x is, therefore, \ x/s, giving 2 nr x/s- 

 as the difference in phase ; with twice this difference for A 2 and B a . 

 The expressions for the five disturbances are thus altered to 



o t 



C COS 2 IT 7p 5 



a x cos 2 7T in -i 1- «i) ; «2 cos 2 tt (~; + — + a 2 J ; 



h COS 2 7T T^ -~ s ~ £l) ; & 2 COS 2 7T ( T - -^ - £ 2 J J 



the signs being chosen on the convention that a positive value of x 

 increases the path to A x . 



Writing <£ for 2 it t / T, all these expressions are of the form 

 F cos ((f) + e) ; and to find their sum we may use the well-known 

 formula (applicable to finding the resultant of coplanar forces at 

 a point) : 



F cos (<f> + e ) + Fj cos ((f) + ei) + F 2 cos (<f> + e 2 ) + . . . . 



= E cos ((f) + E), 

 -ith 



with 



B 2 = F 2 + F x 2 + F, 2 + . . . . + 2 F Fx cos (e - e,) 

 + 2 F F 2 cos (e - e 2 ) + 2 F x F 2 cos (e x - e 2 ) + .... 



