334 
Proceedings of Royal Society of Edinburgh. [sess. 
/ e 2 = ’088 113 ; a 2 = *364 816 ; /3 2 = *452 929^ 
] y 2 = *427 716 ; y 2 - a 2 =*062 900; /3 2 + e 2 = *541 042 l 
^2^2 + e 2 = *993 971 ; 2/3 2 - € 2 = *817 745 ; 4j8 2 - e 2 = 1*723 603 J 
0-058 921 8a? 3 — 0*158 3148^ + 0*097 2308^-0*000 248858 = 0. 
This equation has three roots, only one of which is consistent 
with other conditions not involved in the algebraic expression of 
the problem : that root is 
a =*0025702, 
whence 
v = 2°. . 59'. . 25", 
a result which gives the maximum value of EOS, and therefore the 
minimum of SOs ; or the maximum of - SOs. 
The value of EOS deduced from the above v is 24° . . 40' . . 23", 
while the constant value of EOs is 22° . . 32' . . 59" ; thus leaving 
for SOs only 2° . . 07' . . 24" as the maximum when Os falls 
between OS and the axis. This leaves so small a difference between 
the interior rays which experience total reflection at the surface of 
the balsam, that it is needless to pursue the investigation farther. 
It may be at once held as demonstrated that the best position for 
the plane of cement is beyond the limit which gives a coincidence 
to the two interior rays OS, Os ; and that we must seek for the best 
possible position beyond that limit. 
The maximum value of EOs - EOS must, since EOs is constant, 
accompany the minimum value of EOS, which minimum value, 
since both angles must always lie on the same side of OR, is zero. 
We might, therefore, be led to suppose that the best value of LOE 
is (v) = 57° . . 55'. But in reality, any value of LOE between 
57°.. 55' and its supplement 122°.. 05' is accompanied by this 
circumstance, that no pencil of extraordinary light is intercepted by 
the balsam. Hence in considering the values of v between these 
limits, we have only to examine the condition of total reflection of 
the ordinary ray; this examination is a matter of comparative 
facility. 
Having thus found a wide range of angles accompanied by no 
interruption of the extraordinary ray, we might enquire what 
particular angle would give the most extensive field of view ; but 
