206 On Achromatic Interference-Bands. 



of the rings viewed through a prism, 



n= £±Mi±goil±v (58) 



The equation of the achromatic curve is then, by (54) , 



• {f+/(\)-V'(W+^=W'W-#, • ( 59 ) 



which represents a circle, whose centre is situated upon the 

 axis of f . 



If the glasses are in contact (a = 0), the locus is certainly 

 real, and passes through the point 



f+/(\ )=0, V = 0; 



that is, the image with rays of wave-length \ of the point of 

 contact (^ = 0, y = 0). The radius of the circle is ^/'(Xq), 

 and increases with the dispersive power of the prism. The 

 other point where the circle meets the axis, 



x = 2X /'(X„), y=0, 



marks the place where the bands, being parallel to the achro- 

 matic curve, attain a special development. It is that which we 

 should have found by an investigation in which the curvature 

 of the band-systems is ignored. 



If a be supposed to increase from zero, other conditions 

 remaining unaltered, the radius of the achromatic circle 

 decreases, while the centre maintains its position. The two 

 places where the circle crosses the axis are thus upon the 

 same side of the image of #=0, y=0. When a is such that 



4 = V{/'WP, (60) 



the circle shrinks into a point, whose situation is defined by 



x = f +/(V> = V'(\>)> y = v = o. . . (61) 



Since there are two coincident achromatic points upon the 

 axis, the condition is satisfied for an achromatic system. By 

 (60), (61), 



so that 



a/b = a 2 , 



t= a + bx 2 = 2a (62) 



This is the same result as was found before (42) by the 

 simpler treatment of the question in which points along the 

 axis were alone considered. 



If a exceed the value specified in (60), the achromatic 

 curve becomes wholly imaginary*. 



* Compare Mascart, Traite d'Optique, t. i. p. 435. 



