386 BIOT ON THE EMPLOYMENT OF POLARIZED LIGHT 
which gives 
b 
[el 8 ae 
and consequently 
{[4]—a} {6 +c} = —4. 
When we have determined the new constants a, 4, c, we im- 
mediately deduce A, B, C from their relations with them. I 
have merely changed C into c to complete the analogy of the 
new notation. 
Let us take now, in the series of one system of solutions, 
three pairs of simultaneous values [«],, 8, ; [«]o, 823 (#3, By, for 
each of which we have determined, from observation, the value 
of [«] corresponding to that of 6. These pairs ought to satisfy 
the hyperbolic relation. We shall therefore have 
{[#],—a}{6, +c} = —4; 
{[a], — a} {B,-+ ce} = — 4; 
{[a]s — a} {83 + c} = — 4. 
And since the first members are equal to the same quantity—4, 
we have, on equating them, 
{[a]o— a} {6. + c}={[e], — a} {6, + c}; 
{[2], — a} {8, + ch ={ [a], — a} {8, +c}. 
On developing the multiplications indicated, the product ac dis- 
appears from each of these equations; and it is in this that the 
simplification consists. Then there remains 
O = [a]38.— [#]1 8, + {[#]e— [#]i}e — (Bo — By) as 
0= [4],8; — [2], 6, + {[4]s — [#h}e— (@s—B)a 
All is now known in these two equations, except the constants 
a and ¢ which occur in the linear form. They may therefore be 
easily deduced by elimination when all the terms associated with 
them have been converted into numbers. a and c¢ being thus 
obtained, we have d by any one of the three equations expressing 
it, and we may thus verify the accuracy of the calculations by 
their agreement. And thence we lastly deduce the values of the 
primitive constants A, B, C, which will be 
b b 
A=a——, B=-— 
. C=G@ 
c 
Such is the simplest and safest method to calculate them accu- 
rately. 
