326 Proceedings of Royal Society of Edinburgh. [sess. 
planes until they meet in the line P ; the plane passing through 
this line P and the point O is parallel to the required refracting 
surface. For the determination then of the angle POM, which is 
supplementary to ADC of fig. 1, we only require the operations of 
ordinary trigonometry. Taking the inclination of the faces of 
calcareous spar at 105°. . 05', as determined by the observations of 
Wollaston and Malus, we find the angle LOM, which the axis 
makes with the arete, to he 63° . . 44' . . 46", and not as Malus has 
it, 66° . . 44' . . 46". 
Log cot 52° .. 32' .. 30" =9*884 3264 
Log J3 = *238 5606 
Log cos 63°.. 44'.. 45"| = 9-645 7658 
This error of three degrees committed by M. Malus seems to have 
run throughout his work, and thus throws considerable uncertainty 
on his determination of the refractive indices. 
Denoting this angle LOM by X, and the semi-axes '604, '673 by 
a, /3, the equation of the plane MP is 
xx M + yy u =l , 
or 
x cos X + y sin X = 1 (MP) ; 
while that of the plane NP is 
, yy N 
+ 
or since 
= 
a/3 cos X 
P 2 
J(a 2 sin A. 2 + /3 2 cos A 2 ) 5 J(a 2 sin A 2 + /3 2 cos A 2 ) 
= 1 , 
a/3 sin A 
x cos A ( y sin A 
a P 
/3 2 J ,J(a 2 sin A 2 + /3 2 cos A 2 ) 
= i; 
that i 
is, 
x cos A t y sin A J(a 2 sin A 2 + ft 2 cos A 2 ) 
+ = ~~ a/3 
• (NP). 
Hence, as the line P is common to both of these planes, we obtain 
by elimination 
. g — a ^/(a 2 sin X 2 + /3 2 cos X 2 ) 
y p sm A — p _ a 2 
. -a + B J( a 2 sin A 2 + /3 2 cos A 2 ) 
x P cos A = a /3% _ " 2 — 
