Interference Methods to Spectroscopic Measurements. 343 



Putting (z — b) for #<0, and z-\-b for x>0, ^i(z) for 

 <t>i(z—b), and ^r 2 ( z ) f° r ^(z + ^Oj ^1 f° r jVOO cos 27rD-r . dz, 

 and Z 2 forJ^ 2 (^)cos27rD^.^, P, for fa{z)dg, and P 2 for 

 J^iW^ we have 



v 2 _ Z 1 2 + Z 2 2 + 2Z 1 Z 2 cos 27rn u 

 11 ~ (Pi + P.) a 



But V x = ~ , and Y 2 = p 2 - ; hence 



V 2_ Pl'V|* + P,'V,' + 2^^ COS TTttu 

 (P 1 + P 2 ) 2 



If n n is small, that is, if the centres of the two sources are 

 close together, 



_ P 1 Y, + P,V, 

 Vu ~ P 1 + P 2 ' 

 and if there are several sources whose centres nearly coincide, 



?P V 



tt — x n y n 



Vn ~"^P~" ' 



If the two sources are alike, excepting a constant factor h, 

 which may represent intensity, 



V 2 __ ^ 2 + ^i 2 + 2AAi cos &a 2 y2 

 Vll= (A + AO 2 ~ * 



Thus, if h = h v 



V n 2 = cos 2 7rn ll . V 2 ; 

 and, generally, if h = ?nh, 



v 2 _ m 2 + 1 + 2m cos &a V9 



If the two sources are small when compared with their 

 distance, V 2 =l. 



VIII. Multiple Source. 



If the sources are all equal, equidistant, and symmetrical, and 



/<i= A, where A 1 is the common interval, and n= — where 



^o 



A is the extreme interval, then 



v _ v 7ij sin 7T/i 



y m — ' —, • 



n Sill TTRx 



Thus if 6(cc) = cos 2 7r — , 

 rv a 



Sill 7T71! 



7T71! 1 — f^ 2 " 



2 B 2 



