- 
346 Demonstration of the Binomial Theorem for [May, 
course their specitic gravity.is the mean of that of the, constituents. 
Piydriodic ‘Acid; thuriatic acid, and hydro-cyanic acid, are (com- 
posed each of one volume of hydrogen united to one volume of 
iodine, chlorine, and cyanogen, respectively ; so that the specific 
gravity of each is a mean of the two substances of which jit is com- 
posed. Nitrous gas is composed of two atoms of oxygen .and one 
of azote. Hence its specific gravity is a mean of that.of twice the 
specific gravity of oxygen + the specific gravity of azote. Am- 
monia is composed of three volumes of hydrogen and one volume 
of azote condensed into two volumes ; so that its specific gravity = 
3 x 70625 + 0-875 e ts Sh t3 : ; ' 
= 0°53125. 
J have omitted euchlorine in the preceding enumeration, because 
it presents an anomaly. Its specific gravity is 2°44, or (supposing 
the specific gravity of oxygen to be 1) 2:196. Now the weight of 
its atom is obtained by multiplying this specific gravity by 213; for 
2:196 x 2°5 = 5°480; and the weight of an atom of jit, sup- 
posing it composed of one atom chlorine (4:498) and one atom 
oxygen (1), is 5°498. Jf this fractional factor continue, after the 
nature of chlorine,has been determined with more rigour than could 
-be expected from the original experiments of Davy, it will show 
that the ratio between the specific gravity of gaseous bodies and the 
weight of their atoms, is not always quite so simple as it seems to 
be from the preceding tables; but the determination of this point 
must be left to future experimenters. . 
ArTIcLE III. 
‘Demonstration of the Binomial Theorem for Fractional and Nega- 
tive Exponents. By Dr. ***. 
Tue binomial theorem requires, according to the nature of the 
exponent, different demonstrations.. In the case of the exponent 
being.an entire number, we have an expansion consisting of a finite 
number of terms; whereas in the other cases, of its being either a 
negative quantity, or a fraction, the expansion consists of an infinite 
number of terms. ‘The first case may be satisfactorily proved by 
the theory of combinations, as James Bernouilli has done 3 or by 
showing the general truth of the law by successive multiplications. 
But for the other cases these methods entirely fail ; and the demon- 
strations that are usually given of the law of expansion in these 
cases are far from being complete. Some have derived the general 
demoiistration from “the theory of fluxions.’ ‘Without examining 
whiether the theory of fluxions can be proved without the assistance 
of this theorem, we shall remark that'so elementary and important 
a’ problem’ Ought, if possible, to be proved before the theory of 
