366 BIOT ON THE EMPLOYMENT OF POLARIZED LIGHT — 
rupt the course of reasoning, I shall suppose this determination 
effected, and shall content myself with relating the evaluations 
which result in the cases employed as examples. 
53. But before passing to these particular applications, it is 
desirable to examine, by analytical considerations, under what 
conditions such a hyperbola can be substituted, as an expres- 
sion of the experiments, for the rectilinear form which repro- 
duces them so approximately ; that is to say, we must charac- 
terize generally the relations of magnitude and of sign which the 
values of the coefficients A, B, C ought to have amongst them in 
order to produce the expected equivalence. For if these con- 
siderations are realized, when adapted to the observations, the 
equivalence in question will become more surely admissible than 
if it could be attributed to an accident of numbers, 
In the first place, if we apply the hyperbola to temperatures 
low enough for the coefficient (A) of the right line to be nega- 
tive, the coefficient A must present itself also with the same sign 
as (A), and, moreover, with an equal or slightly different value ; 
for the two laws coincide analytically when e is null, so that 
they should give identically, for this case, the same value of 
[«] if the observations were absolutely exact. Consequently, the 
two coefficients A and (A) ought not to differ from one an- 
other, as regards sign and value, except within the limits of error 
of the results employed for their determination, owing to the 
different manner in which these results enter into them. 
Secondly, I assert that the coefficient C should be a number 
large enough, and such as that the relation z differs very little 
from the primitive coefficient (B), both as to its sign and ab- 
solute value. For if this condition is fulfilled, since e is always 
a fraction less than 1 in the physical applications, the factor 
= o although analytically variable, will always have values 
nearly constant in the particular cases of experiment to which 
it is applied ; which, joined to the essentially fractional character 
Be 
e+C 
or even sensibly proportional to e, as the rectilinear approxi- 
mation presented it. 
54. Now, to show with what accuracy all these considerations 
of e, will always give a product very nearly proportional, 
