518 Prof. Arthur Schuster on 



changing the variable to ys = x 7 the integral (1) becomes 



Acos/3f+^ a „ „ 



1 cos 27rx/\f(at — x)ax. ... (2) 



For a given position of collimator and telescope, it is known 

 that a homogeneous ray of a certain wave-length will have 

 its principal maximum of the first-order spectrum at the focus 

 of the telescope, and it is easily seen that the wave-length in 

 question is given by the same relation by means of which we 

 have defined A. Also if y l =f(at) is the disturbance at any 

 point of the incident beam, y=f(at — x) will be the disturbance 



Fig. 3. 



at a distance x, the wave travelling in the positive direction. 

 If, therefore, in fig. 3 the thick line represents the shape of a 

 wave travelling from left to right, and the thin line is the 

 cosine curve y 2 = cos 2irxj\ ) having as many periods as there 

 are lines on the grating, the disturbance at the focus of the 

 telescope is proportional to J y x y 2 dx. 



The disturbance at all times is obtained by letting the wave 

 travel forward, the cosine curve remaining in the same 

 position. 



9. We shall use equation (1) as the basis of our calculations. 

 It is required to find the disturbance at F (fig. 1) for any 

 given position of the telescope. We may define the position 

 either by the quantity X or by q = 2iry(\ ; when convenient we 

 shall introduce the number of lines on the grating 2N=ql/7r. 



As a first example, let f(at) =p cos pt, that is to say, let the 

 incident beam be homogeneous, the maximum displacement 

 being unity. Leaving out constant factors, the integral to be 

 determined is 



p cos qscos(pt—fcs)ds, 



J: 



where k is written shortly for py/a. 



Remembering that ql is a multiple of 2ir, the integral is 

 easily seen to become 



g qp 2 sin kI cos pt (3) 



q ~~ k 



