Interference Phenomena. 517 



In fig. 2 let a plane-wave front LM advance towards a 

 grating AB having its centre at C, and let the disturbance at 

 some point of the line PC be given as/(a£), t being the time 

 and a the velocity of light. By a proper choice of the origin 

 of time we may express the disturbance at any other point 

 by the same function. If we imagine a telescope pointed in 

 the direction OC, the disturbance at the focus of the object- 

 glass will depend on the disturbance at some previous time 

 over the different points on the line HK at right angles to 

 CO, account being taken only of that part of the disturbance 

 which reaches a point F on HK in a direction DF parallel to 

 CO. Let F' be a point on CD such that DF = DF, and 

 assume that the disturbance at any time which reaches F 

 from D is the same as that which would have reached F' if 

 the grating had been absent, the amount only being reduced 

 in the ratio cos qs to unity. The excess of optical length 

 GDF over PCO is s (sin ft — sin a) , where a and ft are the 

 angles formed by the normal to the grating with CP and CO 

 respectively. Hence if s is measured from C and /(at) is the 

 disturbance at 0, the disturbance at F will be 



cos qsf{at— s(sin/3— sin a)}. 



The amplitude at the focus of the telescope will be propor- 

 tional to the above expression integrated over the effective 

 aperture. If h be the length of the lines ruled on the gra- 

 ting, 21 its width, the integral will become, writing y for 

 sin/3— sin a, 



h cos ft I cosgsf(at — <ys)ds (1) 



It is of course immaterial whether the beam is limited by the 

 grating, or whether we have an infinite grating, and limit 

 the beam by covering the object-glass by a diaphragm of 

 length h and breadth 2 I cos /3. 



A more rigorous analysis which is given further on (22) 

 will show that the expression which has been deduced when 

 multiplied by a constant factor correctly represents the 

 amplitude of the disturbance at the focus of the telescope, if 

 the origin of time is properly chosen, and if /(at) expresses 

 the velocity of displacement in the incident beam. The 

 factor by which (1) must be multiplied in order to give 

 correct numerical values is l/27raF, where a is the velocity of 

 light and F the focal length of the telescope. 



We get a clear idea of the meaning of the integral (1) by 

 its geometrical interpretation. The distance between suc- 

 cessive lines of the grating is 27r/q, and if there are 2N lines 

 on the grating it follows that <// = 27rN ; writing \= 27r<y/q and 



