368 Proceedings of Boyal Society of Edinburgh. [june i, 
of achromatism for the particular values of the refractive indices 
from the quantities F and A . As this equation is given in all 
optical treatises, it is unnecessary to introduce it here. I suppose 
the computation made, and the four quantities F^, A, f-^ and f^ 
determined. 
I shall continue to use the notation of the preceding section, the 
four surfaces with their relative lines and angles being distinguished 
as before by numerical suffixes. The suffixes 1 and 2 apply to the 
outer and inner surfaces of the crown lens ; the suffixes 3 and 4 
apply to the outer and inner surfaces of the flint lens, or corrector. 
(1) It is analytically requisite that each lens shall he aplanatic; 
that is, each lens is to bring the rays which are incident on it to a 
true focus after two refractions, otherwise the means of comparing 
the refractions of the two lenses would not exist. 
(2) The lenses must he aplanatic in combination; or, the rays 
after four refractions are to converge to a true focus. 
(3) As the focal lengths of the two lenses are determined by the 
achromatic equation of condition, the aplanatic conditions (1) and (2) 
of this paragraph must be independent of the focal lengths. This 
last-mentioned condition can only he satisfied if the curves (2) and 
(3) have a common pole; because then only may the variable 
surface (3) of the second lens be moved to any suitable position 
between the pole and the first lens, and yet every ray of the 
converging cone coincide with a radius vector of curve (2), and also 
coincide with a radius vector of curve (3). This implies that the 
curves (2) and (3) are to he similar and confocal curves. 
(4) There are four curves, and four parameters, «i 234 > 
achromatic equation of condition only involves one relation amongst 
them. The parameters, are already determined by the condition 
that they are to have a common pole, which implies that are to 
he equal to the distances of the lenses from that pole. We may 
further determine «j = a 2 , which implies that the two surfaces of 
the crown lens are of nearly equal convexity., the first being of the 
form, /(cos n-fi^, and the second being of the form, /(sec n^6(^ . is 
then to he determined consistently with the achromatic equation of 
condition. The same angular coordinate, Oq, expresses the position 
of corresponding points of surfaces 1, 2, and 3, referred to the 
common pole as origin. 
