367 
1888.] Lord McLaren on an Aplanatic Ohjective. 
Proceeding as in paragraph (1), we find 
9 6 
Also, Wj + (02 = </>i' + ff> 2 ' Hence, by substitution, 
4" ~ (^1 + 1 4" n ,^9 . 
The condition of minimum deviation, = gives 7?2 = % + L 
and the last equation reduces to the two forms 
(2?1i 4- l)/x = 2?^J4-2 : ( 2 ti 2 - 1)/^ = 2 ?^ 2 , whence 
_ 2yzi 4- 2 
. 
2/x-2^ 
^0 = 
^ 27^l4-l~27^2-l^ 
For the value /x = 1 '5, J : ?Z 2 = 3/2. 
The first surface is = a^cos {\9) (The cardioid) 
The second surface is . sec^^ 
2 2/X-2' 
)• 
(13). 
(14). 
The inclination of the asymptotes of the second surface is. 
?« = |x^ = 60 ». 
To find the Surfaces of an Aplanatic and Achromatic Combina- 
tion with minimum Deviation of the Ray. 
(1.) Where the Lenses are not in Contact. {The Dialyte.) 
In this form of the telescope, the crown lens only is at the object- 
end of the instrument, and in the cone of converging rays a smaller 
flint lens is placed, having the focal length which is requisite for 
the correction of the chromatic error (see figure 9). 
The preceding analysis, when applied to the determination of the 
surfaces of an instrument of this construction, gives results which 
are at once simple and symmetrical. 
The following data are supposed to be given : — 
(1) The principal focal length F., which depends of course on 
the size of telescope wanted. 
(2) The distance between the centres of the crown and flint 
lenses, which may be denoted by A. This depends evidently on 
the relative diameters of the crown and flint lenses, and may he 
determined arbitrarily. 
(3) The focal lengths /^, of the crown and flint lenses : These 
are to he determined by the known formula, or equation of condition 
VOL. XV. 31/10/88 2 A 
