gold had been thus separated, by exposing it to the action of nitromuriatic acid, and afterwards evaporating the excess of acid and adding distilled water, it was found to contain gold, which was thrown down, from the muriate thus formed, by the usual tests of sulphate of iron and muriate of tin. The most curious thing to be attended to in separating the portion of metallic gold by means of mercury, in the foregoing experiment, is this; that the success of it depends upon the moist state of the precipitate after being washed with water; but whether the revival of the gold be owing to the hydrogen of the water, or to the cause assigned by Berzelius, of the spontaneous decomposition of the protoxide of the metal*, others may determine. Twelve grains of the purple powder, which had been previously exposed to a red heat in a porcelain capsule, were agitated with well dried mercury; but, after the volatilization of the mercury, no film of gold remained, as in the former instance; nor any residue, whatsoever, to alter the weight of the watch-glass in which this trial of the dry powder had been made. The same dry powder, after the removal of the mercury, being placed in nitro-muriatic acid, and the excess of acid expelled, and distilled water added, yielded a solution, from which gold was precipitated by the usual tests.

From all the preceding observations it may be inferred that in precipitating the purple powder of Cassius from the muriate of gold, by means of the muriate of tin, the two metals tin and gold are thrown down as oxides; which however do not chemically combine in a constant relative proportion to each other; that the quantity of tin always exceeds that of the gold; and that the difference observable in the hues of the precipitate made at different times, is to be ascribed to the different proportions in which the oxides of the two metals have combined together, and perhaps also to their different degrees of oxidation.

EDWARD DANIEL CLARKE.

* See a former Note.

V. Observations on the Notation employed in the Calculus

of Functions.

BY CHARLES BABBAGE, M. A.

FELLOW OF THE ROYAL SOCIETIES OF LONDON AND EDINBURGH,

AND OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY.

[Read May 1, 1820.]

AMONGST the various causes which combine in enabling us by the use of analytical reasoning to connect through a long succession of intermediate steps the data of a question with its solution, no one exerts a more powerful influence than the brevity and compactness which is so peculiar to the language employed. The progress of improvement in leading us from the simpler up to the most complex relations has gradually produced new modes of shortening the ancient paths, and the symbols which have thus been invented in many instances from a partial view, or for very limited purposes, have themselves given rise to questions far beyond the expectations of their authors, and which have materially contributed to the progress of the science. Few indeed have been so fortunate as at once to perceive all the bearings and foresee all the consequences which result either necessarily or analogically even from some of the simplest improvements.

The first analyst who employed the very natural abbreviation of instead of aa little contemplated the existence of fractional negative and imaginary exponents, at the moment when he

adopted this apparently insignificant mode of abridging his labor. So great however is the connection that subsists between all branches of pure analysis, that we cannot employ a new symbol or make a new definition, without at once introducing a whole train of consequences, and in defiance of ourselves, the very sign we have created, and on which we have bestowed a meaning, itself almost prescribes the path our future investigations are to follow.

Such being the power and influence of those symbols* by which mathematical reasoning is carried on, it cannot be considered as unimportant either as regards the particular branch, or with reference to the science in general, to examine some of the bearings of the notation which has been employed in the calculus of functions, and to resolve some of the unusual questions which it presents.

That the results to which such an enquiry will conduct us are of a nature purely speculative, is an objection to which every attempt to improve notation is liable; it can however never be considered an useless task to examine and strengthen that which essentially contributes to the power of an instrument, which enables us so wonderfully to trace the connexion between the phenomena of nature.

When it became convenient to express without performing the repetition of an operation, whose characteristic is f, the method which first presented itself doubtless was similar to that which had been adopted with such advantage for the exponents of quantities, and f(x) was written instead of ff(x); it now followed without any other convention that f(x) and f" (x) represented fff. (x), and fff. (n times) (x) and also that

ƒn+m(x) = ƒnƒm (x).

when n and m are whole numbers.

(4),

* Euler, to whom analysis is so much indebted, appears to have been fully aware of the power and importance of notation; "Summa analyseos inventa maximam partem algorithme ad certas quasdem quantitates accommodato innitantur." Specimen Algorithmi singularis. Acad. Pétrop. Comm. Nov. 1762.

At this point of generalization a question occurred as to the meaning of f" when n is a fractional, surd, or negative number, and in order to determine it, recourse was had to a new convention not inconsistent with, but comprehending in it the former one. The index n was now defined by means of the equation (4) and was said to indicate such a modification of the function to which it is attached that that equation shall be verified.

From this extended view of the equation (A) several curious results follow; if n=0, it becomes

This informs us that fo is such an operation that when performed on any quantity it does not change it, or putting fTM (x)=y, it gives

ƒ° (y) = y,

a result which is analogous to x=1.

Let m=1, n=1, we have

f'x=f'f'(x), orƒ(ƒ−1x) = x;

f-1(x) must therefore signify such a function of x, that if we perform upon it the operation denoted by f it shall be reduced to x. The number of functions possessing this property will depend on the nature of f; thus if f(x)=x", let r1, r,,....r, be the roots of "-1=0; then f(x) may be either of the n quantities

for if we perform the operation fƒ upon any of them, or raise it to the nth power, the result is x.

Here then we find

ƒ(ƒ−1x) = x',

in all cases; but if the negative index is attached to the first functional sign, we have