356 
Proceedings of Royal Society of Edinhurgh. [june 4 , 
satisfy the condition of achromatism; and a relation between the 
three factors, is to be found consistent with the required 
condition of minimum deviation. The quantities, have no 
direct relation to the focal lengths ; nor have the quantities, 71^23, 
any direct relation to the six sines of incidence and refraction, 
^123 ^^123 > their products. The problem is therefore extremely 
complicated; and a direct and rigorous solution seems to be un- 
attainable by known methods. 
It will be shown, however, that by assuming an approximate law 
of refraction, or approximate refractive index, 
( 
instead of 
_ sin 
curves can be found which are sensibly aplanatic and achromatic 
for lenses of the usual apertures, and whose elements can be 
determined for any given focal length and aperture, so as to 
satisfy conditions ( 1 ), (2), and ( 3 ) rigorously. One of these surfaces 
may be a plane or spherical surface, which complicates the theory, 
but simplifies the practical conditions of the case. 
It is an interesting and practically important result of this 
analysis, that if spherical surfaces only are required, such as near 
the centre of the lenses will satisfy the three conditions of aplanatisni 
and achromatism, the analysis determines the radii of the four lens 
surfaces with absolute accuracy. 
Because, if we compare the curve of approximate aplanatism, which 
may be denoted by / , with the absolute aplanatic curve 
^ common vertex, we have the ratio ^ 
ultimately = . Consequently at the vertex the curves 
coincide, having a contact of the second or some higher order. 
Their curvature at the vertex is therefore the same. In the 
spherical system, we have then only to take for the radii of the four 
spherical surfaces, the corresponding radii of curvature of the 
aplanatic surfaces. It will be shown that these radii of curvature 
are very easily found for curves of the type f(^^ ' 
The investigation of these approximately aplanatic curves is the 
