370 Proceedings of Royal Society of Edinhurgh. [june 4, 
first surface is of the form, = a” cos nO^ and the equations of the 
second, third, and fourth surfaces are of the form, r” = a”sec?i0. 
The four values of a and n are 
The equations for lenses in contact, with parameters are 
immediately derived from these by making and 
(2.) Achromatic Aplanatic Combination ; Lenses in Contact. 
As there are six constants, 72 ^ 23 , and as the two conditions 
of achromatism are satisfied by one relation amongst the constants, 
the problem will be indeterminate unless two of the constants are 
determined arbitrarily. This may be done in various ways. 
(1) The surfaces may be determined under the condition that two 
of them, say the two surfaces of the double convex, shall be equal. 
This involves the grinding the four surfaces to aplanatic curvature, 
and while there may be more trouble in figuring four curved surfaces 
than in working to a design which contains only three curved 
surfaces and a plane, I am disposed to think that the superiority of 
this form of lens in point of analytic simplicity to the form which 
is hereafter investigated, points to this as being also practically the 
more perfect form of lens. The equations for this surface have 
been already found from the formulae for lenses not in contact. 
(2) One of the surfaces may be determined arbitrarily, and it 
may be either a circle, {r = a cos or a plane, {r = a sec 6-f, accord- 
ing to the surface selected. But this cannot be the intermediate 
surface, because the condition of minimum deviation makes it 
necessary that a relation be found between the constants n, rq of two 
■ ( 15 ). 
(16). 
(Eq. 16). 
