fluxions is established; and however ingenious these demonstrations may be, they cannot be admitted as the proper demonstrations by which the truth is first to be established. To the common demonstrations given in elementary books, we object that they are not general. The law of the coefficients is shown for the first coeff

cients, and this induction is gene without any solid demon

1

stration. (Nov. Comm. Petr. vol. xix. p. 103; Phil. Trans. 1806, p. 318.) A demonstration for those cases, which is at once satisfactory and elementary, seems, therefore, not to be generally known, and may be very desirable. We intend to submit to the judgment of the reader one which appears to us to have these characteristics, and is, as far as we know, new. We shall suppose that it has been proved that n, being an entire number,

· (a + b)" = a2 + n a2→ 1 b +

n-r+ 1 a"-" }, &c.; and shall prove that, n being

3

any number, the same expansion is true.

For the sake of abridgment, we shall denote the binomial coeffithus "I", where the number on

cients

n.n T.n- 2...nr + 1

1.2.3

r

the right denotes the number of factors both in the numerator and denominator, and the number on the left expresses the first factor, n, or the exponent of the binomial quantity. We have, therefore, generally, the following relation between two such coefficients :-± ✦✦1[" or "I" n − r = r f 1+1 (8) whatever number n may be, and the theorem for n an entire number, is thus expressed :—

(a)

In `n - r
r+1

(a + b)" = a2 + 1In a2-1b + 2【n an - 2 b+ and I am ris &c. + bn.

We shall now demonstrate that, whatever p or q may be, we have always

For r = 1, it will be easily seen that 'I + 9 = ' + 'Io.

We say that the demonstration is new, because we believe the demonstration in that general form in which we have given it to be new. Euler's demonstration in Novi Comment. Acad. Petrop. 19, is indeed much the same, as far as it goes ; but Euler shows only the form of the first two coefficients, and says, Quemadmodum hic duos primos coefficientes per literas met n determinare licebat, ita manifestum est, si superior multiplicatio ulterius continuaretur inde etiam sequentes coefficientes C, D, E, per easdem literas m et n definiri posse quamvis calculus mox ita foret molestus ut maximum laborem requireret. It is evident that the agreeing of the first two coefficients with the same coefficients for entire numbers, which is Euler's demonstration, cannot be satisfactory and strict. The same objection applies with equal force to Dr. Robertson's demonstration, which seems nearly to agree with that of Euler in the paper above referred to..

2

,,

·4] + 9 = +[P + sp; 1]? + [p 2]? + '] [?: + 418.

But in order to generalize this, induction, and to show that this must always be so, we shall suppose that, proceeding in this manner, we had convinced ourselves of its truth up to the values of r, so that we had proved that

·3]p + 9 = 3 3 + - !IP 119 +
sIp
And we shall prove that it

·$ + !]$ + 9 = 3] +9

2]3 ?I? + - 3] 3], &c..

will also be true for rs+ 1; for

p+q S

s+ 1

r + 1

, and writing the latter

s + 2) + (q

s + 1

2)

1) (p

&c. we

obtain

+

31

+ !

The first term is

+ 'I (a). We have also

-2IP. 219 p. - s + 2
s+ 1

s + 1

+

s+ 1

$ + 1

(B)

And thus two terms produced by two successive terms will always give together the same terms in the product. Thus in the pro

ducts

two parts give together +-'].

And it follows that the whole product will be

which is of the same form. If, therefore, the proportion be true for rs, it will also be true for rs + 1. But as we proved it to be true for s= 1, s = 2, s = 3, it follows hence that it will be true for all succeeding values of r, or that it will be generally

true.

Let us next assume two quantities of this form :

Where p and q may be any positive numbers, and multiply them together, the product will be as follows

:

It will now be readily seen that the coefficient of a2 + q = b2 n = *I*+, and that the whole product will be

QP + 9 + 1IP + 9 a2 + q = 1 b + ?I? + 9 a2 + q −2 b2, &c. + *[p+

ap + q➡+ b3, &c. which, being still of the same form as the factors which produced it, gives, if multiplied by another quantity of the same form,

-1

-2

a2 + 1I at − 1 b + Ir a22 b2, &c. the product

a2 + q + r + 1Įp + q + r ap + q + r = 1 b + ? [ l + 9 + * ap + q+r=2 l2, &c. which is still of the same form. This being multiplied again by a quantity of the same form, would produce a quantity of a similar form. Let us now suppose that all the p, q, r, &c. are equal, let that number = n, and the sum npm an entire number, so that p, and all those quantities which are multiplied will be equal, and we shall have for their product

=

n

I amp- 1o, &c. and consequently patting for np, m; and for

a + 1 @ → b + 3Ï" @" ~2 1o, &c. = "/ (am + 1Ịm, qm −1 b + 21 am-2 f2 + &c.'

but by the binomial theorem for entire numbers, the part on the left under the radical sign is (a+b), and therefore f

1

m

=

-2

q + I a = b + 2I" a" - 2 b2; &c. = W (a + b)" =

312

which is the binomial formula for fractional exponents, As

4

= (aP + 9 + " + 3[k+?+* @£+9+d=r b 4 [p + 9 + r ap +4 +r→ 2 f2; &c.

we have, putting p+q+r=w, and p + q = w — r,

This equation will be true for any value of r and w, and consequently also for wo, by which we have

=

1

= (a + b)", and

a2 + 1]" @" - 1 b + @Ir ar ~ b2, &c. (a + b) r consequently the truth of the theorem is also proved for any negative number.

ARTICLE IV.

Experiments on Prussic Acid. By M. Gay-Lussac.t

(Presented to the Institute, Sept. 18, 1815.)

THE experiments which I have the honour to communicate to the Class have for their object the nature of prussic acid and its combinations. Few bodies have been more studied, and yet few are less known. After the labours of Macquer, Scheele, and Berthollet, which form an epoch in the history of prussic acid, many other distinguished chemists made experiments upon it. I shall not however attempt to write a history of these; but merely notice the principal results which they have furnished, in order to point out the place from which I started.

The case for a fractional exponent having already been proved. + Translated from the Ann. de Chim, vol, xcv. p. 136.

It is to Macquer that we owe the first important experiments on Prussian blue. On boiling it with a solution of potash in excess, that skilful chemist observed that nothing remained but oxide of iron, while the alkali combined with the colouring matter. On the other hand, if the Prussian blue predominate, the potash is com pletely saturated with the colouring matter, and loses its alkaline properties, just as if it were saturated with an acid. In both cases it has acquired the property of producing Prussian blue with solutions of iron, by means of double affinity; and it precipitates the greater number of the other metalline solutions. When Prussian blue is calcined, volatile alkali is formed-a fact which had likewise been observed by Geoffroy-and there remains oxide of iron attracted by the magnet, and a quantity of charcoal. From these experiments, Macquer concluded that Prussian blue is a compound of oxide of iron with an inflammable substance, which is converted by calcination into volatile alkali and charcoal.

Twenty years after, MM. de Morveau and Bergman considered the colouring matter as a particular acid, and the first gave it the name of prussic acid. However, its true nature remained still

unknown.

Scheele, whose name is connected with so many brilliant discoveries, succeeded in 1782 in obtaining prussic acid in a separate state by a very ingenious process, and approached very near to a knowledge of the true nature of its constituents; for he ascertained that it is obtained by the union of ammonia with a charry matter rendered volatile by heat. †

To these important experiments of Scheele succeeded those of Berthollet, which are not less so. He showed that Macquer's combination of the colouring matter and potash is a triple salt, of which iron constitutes the third element. On mixing chlorine with prussic acid, as obtained by Scheele, he observed that the first substance is changed into muriatic acid, and that the second has acquired a much stronger smell, and has lost part of its affinity for alkaline bases. In this new state it no longer forms Prussian blue with solutions of iron, but a green precipitate, which becomes blue when exposed to light, or when mixed with sulphurous acid. If potash be added, the prussic acid is completely destroyed, ammonia is produced, which is disengaged; and carbonic acid, which remains combined with the potash. From these results, and from the knowledge of the elements of ammonia, for which likewise we are indebted to Berthollet, he considered prussic acid as a compound of carbon, azote, and hydrogen. He does not admit oxygen into the number of its elements, and he supposes that the oxygen contained in the carbonic acid produced by the action of potash on prussic acid altered by chlorine, is furnished by this last substance. The absence of oxygen in prussic acid not being rigorously demon

* Mem. de l'Acad. des Sciences, 1752. + Opusc. de Scheele, vol. ii. p. 141,

1 ↑ Ann, de Chim. vol. i. p. 30.