368 BIOT ON THE EMPLOYMENT OF POLARIZED LIGHT 
All the relations whica we had foreseen between the numerical 
coefficients A, B, C, are here very precisely realized, both as 
regards their signs and their relative magnitudes. For, first, 
the excess of A over (A) is only 0°37539: then, [«] being here 
expressed in degrees in the two laws, there would only result a 
difference of 22' 31" over its evaluation in degrees, for the case 
of e null, if that case could be realized experimentally ; and this 
deduction, which is the furthest removed from the determining 
data, could only be obtained within this limit of certainty by 
reiterated observations. The other extreme evaluation of [«] cor- 
responding to e = 1, presents a still nearer agreement, as less 
distant from these same data; for it is found to be + 13°1355 by 
the rectilinear law, and + 13°°10024 by the hyperbola, which does 
not present any difference physically appreciable. The coinci- 
dence of the intermediate values of [a] is rendered evident if 
their hyperbolic expression be considered in its second form, 
For, in truth, the coefficient of e, 15°°67075, there exceeds a lit- 
tle its analogue of the rectilinear relation, which is only 14°*3154. 
But this latter is only divided by unity in the rectilinear expres- 
sion of [«], whilst the corresponding term of the hyperbola is 
divided by 1 + e. 0°069225, a number always somewhat greater 
than 1; and it is this that produces the exact agreement of the 
two values of [«] at their extreme limit, where e=1. We may 
therefore legitimately consider these two general expressions of 
the phzenomenon as being equally admissible, and numerically 
equivalent within the limits of uncertainty which the observa- 
tions admit; but the hyperbolic law is physically preferable, 
because it is less restricted. And if other phenomena of com- 
bination, analogous to those, but more extended or more ener- 
getic, reproduce it with undoubted curvilinear characters, as in 
fact we shall see further on, there will be the strongest presump- 
tion that it is this which is actually realized, although in a less 
evident manner, in the successive but less intimate combina- 
tions of tartaric acid with water. 
56. Having thus demonstrated these two equivalent modes 
of variation of the function [«] which our experiments realize, 
let us compare them with the various forms which that function 
may assume in the systems composed of an active substance and 
an inactive liquid, forms which we have enumerated and charac- 
terized in § 27 for all the supposable physical states of similar 
