On Images formed without Reflection or Refraction. 215 



tance less than f 1} it will appear magnified. Inasmuch as the 

 arrangement affords a view of the sun with full definition and 

 with an increased apparent magnitude, the name of a telescope 

 can hardly be denied to it. 



As the minimum focal length increases with the square of 

 the aperture, a quite impracticable distance would be required 

 to rival the resolving-power of a modern telescope. Even for 

 an aperture of four inches/! would be five miles. 



A similar argument to that just employed to find at what 

 point a lens begins to have an advantage over a simple aper- 

 ture, may be applied to determine at what point an achromatic 

 lens begins to assert a perceptible superiority over a single 

 lens in forming a white image. The question in any case is 

 simply whether, when the adjustment is correct for the central 

 rays of the spectrum, the error of phase for the most extreme 

 rays (which it is necessary to consider) amounts to a quarter 

 of a wave-length. If not, the substitution of an achromatic 

 lens will be of no advantage. 



If fjb be the refractive index for which the adjustment is 

 perfect, then the error of phase for the ray of index f^ + Bfi is 

 SfjL . t, where t is the " thickness " of the lens. Now 



(/*-!)<= J; 



so that, if the error of phase amount to | A-, 



■V _Vi 

 fju-l~2r 2 ' 



In order to apply this numerically, let us take the case of hard 

 crown-glass, for which the indices are given by Hopkinson*. 

 The practical limits of the spectrum being taken at B and G, we 

 have/z-B — 1*5136, /z G = 1*5284, the difference of which is* 0148. 

 If the focus be correct for the mean value of ja, the extreme value 

 of o> is -0074, and that of o>/t>-l) is -0074/ -521, or -0142. 

 In strictness we ought to take into account the variation of A, ; 

 but for such a purpose as the present we may put it at ^q-Jo o 

 inch; and then the fraction *0142 expresses the admissible focus 

 when a single lens is used as compared with the focus neces- 

 sary when a lens is dispensed with altogether. Thus, if the 

 aperture be one fifth of an inch, an achromatic lens has no 

 advantage over a single one, if the focal length be greater than 

 about 11 inches. If, on the other hand, we suppose the focal 

 length to be 66 feet, a single lens is practically perfect up to 

 an aperture of 1*7 inch. The effect of spherical aberration in 



* Pi-oc. Roy. Soc. 1877. 



