338 Proceedings of Royal Society of Edinburgh. [sess. 
and is affected by the corresponding error. But I do not think 
that Dr Sang’s further remark is justified, as Malus not only gives 
the correct expression for the cosine of the angle in question, but 
seems to have employed in his subsequent calculations the inclina- 
tion of the axis to a face , not to an edge, of the crystal : — and he 
gives the accurate numerical value of this quantity, as deduced from 
Wollaston’s measure of the angle between two faces. 
There is an altogether unnecessarily tedious piece of analysis in 
Dr Sang’s investigation of the limits within which the prism works: — 
and it is so even although he shortens it by the introduction of the 
terribly significant clause “ after repeated simplifications.” I will 
give below what I consider to be a natural and obvious mode of 
dealing with the question (one which, besides, leads to some elegant 
results) : — but I have reproduced Dr Sang’s manuscript as it was read , 
for the circumstances of the present publication seem to require 
literal accuracy. Dr Knott has kindly verified for me the agreement 
of my final equation with that of Dr Sang. 
In p. 331, above, it is clear that, since S is a point on the spheroid, 
we may put 
x s — a cos (fj , y s = /3 sin <j > . 
But we have (p. 328) 
x n — y cos v, 2/r = y sin v . 
Hence the general relation between B and S, i.e., between c fi and v , is 
cos cos v sin <£ sin v 1 
“ 1 t ~ 'V 
Also, since the angle BOS is to be a maximum, 
^(tan-(Aan^)-v) = 0. 
Differentiating the first equation, and eliminating d<fi/dv between the 
two, we get at once the remarkably simple relation 
(tan </>) 3 = — — tan v 
But we may put the first into the form 
(i). 
or 
cos v sin v , , 1 , 
1 — — tan <f> = — sec A , 
a P 7 
(cos I/) 2 1 2cos v sin v 
— + 
7 
. . . , /(sin i/) 2 1 
^ tan <f> + y—jp ^2 
tan</>) 2 = 0 . (2). 
