1888 .] Lord McLaren on an Aplanatic Ohjective. 
355 
4. On the Pour Surfaces of an Aplanatic Objective. 
By the Hon. Lord McLaren (Plates XIY., XV.) 
The art of figuring the lenses of a telescopic objective consists in 
correcting the spherical aberration by successive trials, until 
approximately aplanatic curves are obtained. 
It is desirable that the forms of these curves should be 
investigated in order that measures may be applied to the larger 
lenses to test the accuracy of their curvatures. If only a single re- 
fracting surface were to be considered, such an investigation pre- 
sents no difficulty ; but the determination of the elements of four 
aplanatic surfaces for two lenses of different densities is evidently a 
very complex problem. 
Conditions of the Problem. 
In order that the two lenses of an object-glass, supposed in con- 
tact, may fulfil the requirements of being aplanatic and achromatic, 
three conditions must be satisfied. Treating the two surfaces which 
are in contact as a single surface, there are three surfaces to be con- 
sidered, and the conditions are — (1) Each of the three surfaces is 
to bring the rays of any given colour incident upon it to a true 
focus, or the spherical aberration is to be nil for each surface 
separately. (2) In order that the original chromatic error (and the 
spherical aberration, if any) may be a minimum, the ray within 
each of the lenses is to be inclined at the angle of minimum 
deviation. (3) In order that the chromatic error may be corrected 
for selected colours, the focal lengths of the two lenses is to conform 
to the known equations of condition of achromatism. 
Regarding condition (1), it results from a known differential 
equation, that every aplanatic curve must be of the form + ^ir^ = nc, 
where /a is the refractive index for the relative media, and c is the 
distance between the foci. 
Now, as there are three curve surfaces to be considered, and two 
arbitrary constants n and c in each equation, or six in all, there are 
more disposable constants than are necessary to satisfy the conditions 
(2) and (3) ; consequently (under limitations hereafter considered), 
we may determine one of the surfaces arbitrarily. A relation is 
then to be found between the three focal distances, which will 
