Correction of Telescopic Objectives. 409 



very simply from those of the doublet *. The ot is the 

 same for both forms. Again it will be obvious from earlier 

 remarks that the same coefficients apply to a doublet with 

 the flint component leading as to one with the crown in 

 front. But perhaps more important than either of these is 

 the indication afforded by the magnitude of a of the purpose 

 for which a given combination of glasses will be useful. 

 The value of w always lies between very narrow limits ; yS 

 is always 3inall, being zero for a single lens or for a cemented 

 combination of different glasses of the same refractive index, 

 and a small positive quantity if, as is usual, the component 

 made from glass with the greater dispersive power has the 

 higher refractive index. On the other hand a varies through 

 a wide range of values. For a single lens of refractive 

 index fi its value is fj, 2 /(/JL — l) 2 . In a doublet of the usual 

 cemented type it falls from this value as the difference 

 between the refractive indices of the two glasses increases. 

 The rate of fall increases with the power of the components 

 relative to that of the complete lens. Generally speaking 

 the possibility of obtaining similar corrections with two dif- 

 ferent combinations of glasses depends upon their having 

 approximately equal values of « . For example, the simul- 

 taneous correction of spherical aberration and coma for unit 

 magnification (??i=— 1, M = 0) requires, from equations (1) 

 and (2), « = /3 = 0, and from equations (4) and (5) it follows 

 that it will be necessary for x to be approximately zero 

 since /3 is small. 



This property of a in determining the type of correction 

 that is possible leads to a novel method of designing instru- 

 ments which are built up of a number of separate lenses 

 when each may be regarded as approximately thin. Each 

 lens is assumed to have the same value of -nr — a value about 

 the middle of the possible range is chosen. The {3 of each 

 lens is assumed to be zero, and the conditions to be satisfied 

 then lead to connected series of values of a for the various 

 component lenses. As the types of glass available do not 

 form a continuous series it will not be possible to realize the 

 majority of these series, but a few can usually be selected 

 with very little difficulty in which these simplified conditions 

 are very approximately satisfied. The most favourable case — 

 often determined by the magnitude of the curvatures in- 

 volved — may be adopted for more detailed investigation with 

 corrected values of a , /3 , and ot based upon the glasses 

 selected. This method of calculation is the inverse of that 

 usually employed, the selection of the refractive indices of 

 * Proc. Phys. Soc. vol. xxviii. p. 232. 



