379 
1888 .] Lord McLaren on an Aplanatic Ohjective. 
arcj = ; arc 2 = these arcs are considered equal, 
ct 
when the thickness of the lens is neglected, 6^ = - 0 -^ . 
«2 
/^1 
.^2 = b02 = %6>j ; = = 
«2 «2 1^2 ^2 
The condition of minimum deviation of the ray, gives — 
= ^ = (a). 
’ c/g /^1 w 
The condition of continuity of the ray within the lens, gives — 
o)i + o >2 = < 1^1 - f - 4>2 
= 2c/>/. 
Substituting for <Op wg and </>/ their values, and dividing by 0 ^ — 
a, , T ^ ^(n + A) 
. 7H-1 + J (^-1) = -^ 
«2 /^i 
Substituting for -1 its value, from (a), and dividing by ?^ + 1 , we find — 
y_^ih ^ (^ ~ ^) _ j 
/^1 * n /^i ' 
-i- {n - l)/-t 2 ~ ; 
/^2 
(/5). 
/^2 + /^1 “ 2 
It is always to be remembered that as a piano-spherical surface is 
aplanatic for the index, of refraction, /x=l*5, — a compound lens, 
whose mean density does not differ much from that quantity, may 
be piano-spherical without deviating sensibly from the condition of 
aplanatisin. 
5. Quaternion Notes. By Prof. Tait. 
(a) Prof. Cayley’s paper, which was read at last meeting, re- 
minded me of an old investigation which I gave only in brief 
abstract in our Proceedings for March 21, 1870 (vii. 143). There 
is, unfortunately, a misprint in the chief formula of transformation. 
In fact, we have quite generally, as a matter of quaternion analysis. 
