240 
Mr. Barrow on the rules and principles for 
Here then are four quantities to be determined, and only 
two equations ; so that if the condition of being achromatic 
was the only one, we might have any variety of answers at 
pleasure ; but it is also required that the object-glass shall be 
free from spherical aberration, which is still only a third 
condition ; and therefore, even with this, the question may 
still be considered as admitting of various solutions. But a 
fourth condition may be that the two interior surfaces shall be 
either actually equal, or very nearly so. And this last con- 
dition serves to bring the solution within very narrow limits, 
although it is still not strictly limited, unless we insist upon 
perfect contact surfaces, or some other specific condition. 
Mr. Herschel, in his very elaborate and valuable Paper on 
this subject in the Phil. Trans. Part II. 1821, instead of this 
last condition, has taken another, viz. “ the destruction of 
aberration, not only for parallel rays, but also for rays diverg- 
ing from a point at any finite distance.” 
The resulting equations by the introduction of this condi- 
tion, make the radii of the two interior surfaces nearly equal ; 
but in several cases the convex side is the deeper, and the 
two surfaces therefore ride in the middle, unless separated at 
the edges by paper, or some other substance interposed 
between them, which by many practical opticians is considered 
objectionable.* The contact surfaces are also in this construc- 
tion deeper, and the actual quantity of aberration in either 
lens to be corrected is greater than would otherwise be ne- 
cessary. Moreover, by insisting upon any fourth condition, 
* Mr. Herschel suggests that the best way would be in all cases to frame each 
lens into a separate cell, and to adjust them to each other by screws. In this case, 
of course, it would be indifferent which of the two were the deeper surface. 
