372 Proceedings of Royal Society of Edinhurgh. [june 4, 
Substituting tbe values of these quantities and multiplying by 
(^) 
/^l(% "b ^ 2 ) ~ "b 1) = 2(//,2^2) = /^2^2 “ ^ * 
Again, by substituting this value of n-^ in the first and third 
members of the preceding equation, we find 
A *' i (/^ 2^2 ~ 1 + ^ 2 ) ~ ^/^ 2^2 • 
By alternate substitution the values required are found to be. 
Wo = 
_ 2/^2 - 
— ^ /t'l 
/^i /^2 /^i ~ 2/^2 /^i /^2 “ 2/^2 + 
For the values = 1 * 5 , /X2= 1 * 75 , these expressions reduce to 
(17). 
12 16 
(18). 
7^0 determine the relative Values of a-^a^ and of 
I shall now take account of the refraction at the plane surface 
and determine relative values of these quantities. The focal length 
(F) of the achromatic combination is determined arbitrarily, accord- 
ing to the instrumental requirements ; and from it the focal lengths, 
of the crown and flint lenses are determined by the condition 
of achromatism. 
1 
Hence, in the expression i = i - ^ , the three quantities are 
J i J n 
known. From these data, the focal lengths of the four surfaces are 
to be found, and thence the parameters a-^a^ of the first and inter- 
mediate surfaces. 
As the lens surfaces are only aplanatic in combination, and not 
separately, the focal length for a central ray for any single surface 
will be that of a spherical lens to the circle of curvature at the 
vertex. The radius of curvature is easily found : for curves of the 
dr 
form, r^ = a^ cos nQ. we have the known relation, p = r^ = 7 — ■■ - v . ■ . 
’ ’ ^ ^ dp {n+\)p 
For the first surface this reduces to p^ = j or, at the vertex, 
Pi=- 
1 
^ 1+1 
Similarly for the second surface, we find 
sec n£ 
P 2 =^ 
Uo-l 
; or, at the vertex, p^ = 
1 
