acid produces an alteration of colour, which is sometimes similar in various acids, but which is most commonly different from that produced by the acid diluted with water; secondly, different acids produce their effects in different times; and thirdly, the colour produced by the action of different acids is more or less durable, and it undergoes changes in a shorter or longer time. It is with respect to these variations that we now propose to consider the acids; and to give the reader an opportunity of judging by comparison, we shall successively enumerate the most common acids, and even those which do not possess a very marked action. Concentrated sulphuric acid, or when mixed with three parts of water, immediately gives a bright rose colour to Brazil wood paper, which, by attracting moisture from the air, becomes orange-coloured. When diluted with rather more water, the acid produces a colour which has a shade of yellow; and with 20 or 30 parts of water, it gives in one minute a yellow, or rather a yellowish colour, which very soon fades. Nitric and muriatic acids act nearly in the same way as sulphuric acid, excepting that the yellow colour produced by these acids, when diluted with water, is paler; the rose colour produced by the concentrated nitric acid soon becomes yellow and greyish, and that produced by muriatic acid still sooner becomes of a dirty grey colour. The action of the three acids mentioned differs but little; but it may be employed to distinguish to a certain extent the degree of concentration of these acids. Sulphurous acid gas bleaches the moistened paper perfectly. Hydriodic acid, when concentrated, gives a rose colour, which gradually becomes yellow on the edges, and in a few days entirely yellow. When diluted with water, it gives a fine yellow colour in half a minute, which soon begins to fade; in a few hours, it becomes less evident, and is rather red than yellow. lodic acid immediately gives a pale dirty yellow colour, which remains unchanged. Concentrated fluoric acid, whether pure or combined with silica, gives a bright red colour; when diluted, it acts in a very marked manner; it immediately becomes a fine lemon yellow colour, which disappears in the space of a minute, and soon leaves a greenish grey colour, which, by transmitted light, appears of an olive-green. When the fluoric acid is used in the gaseous state, it is sufficient to subject the moistened Brazil wood paper to its action for a few seconds. The paper is then stained a bright yellow, which disappears in the manner already mentioned. This also takes place with other volatile acids. Fluoboric acid acts in the same way as the fluoric. Boracic acid has no immediate action, but the colour of the paper soon becomes pale, and is eventually of a reddish-white colour. If the boracic acid contains any trace of sulphuric acid, which is always the case when it is not purified by repeated crystallization, it immediately occasions a very distinct yellowish colour, which soon disappears. The boracic acid of the Isle of Volcano acts very distinctly like pure boracic acid. Concentrated phosphoric acid gives a rose colour, by absorbing moisture from the air, it is slowly changed to orange colour. The acid, when diluted with 10 to 30 parts of water, gives in half a minute a very fine yellow colour, which remains without any alteration. Phosphatic acid cannot be distinguished by its action from phosphoric acid. Concentrated phosphorous acid gives a rose colour, which becomes of a yellow colour sooner than either of the last mentioned acids, and it resembles the colour produced by those acids when diluted. Dilute phosphatic (phosphorous?) acid gives a fine yellow colour, which soon becomes pale. Concentrated hypophosphorous acid gives also a red colour, which becomes gradually pale-yellow, and eventually almost colourless; when diluted with water it gives at first a yellow colour nearly as fine as that of the three last named acids, but it soon disappears, and there remains an indistinct colour, which is neither red nor yellow. Concentrated arsenic acid produces a rose colour which remains for a long time. Diluted with 10 to 30 parts of water, it gives in one minute a very fine yellow colour, but in a few minutes, it fades, and becomes and remains pale-yellow. Arsenious acid has no marked action. Concentrated acetic acid gives a dull-yellow colour directly, which disappears immediately, and is succeeded by a paleviolet colour, which, by transmitted light, is of a very deep reddish-violet colour. Diluted with more or less water, it gives at first a yellowish colour, and afterwards, both by transmitted and reflected light, a reddish-violet colour. It is to be observed that the reddish-violet colour does not become very evident for half, or sometimes one hour, and after some hours have elapsed, the tint is still stronger; it then becomes almost as deep as the colour produced by the alkalies. If the acetic acid is not pure; if, for example, it contains sulphurous or sulphuric acid, which sometimes occur in acetic acid as usually prepared, their presence is easily detected by Brazil wood paper. Sulphurous acid destroys the action of acetic acid, or renders it extremely weak, according to the quantity which it contains, and sulphuric acid causes th acetic acid to give a yellowish colour, instead of the reddish violet. By this method, very small quantities of sulphuric acid may be discovered; acetic acid, for instance, which contains only 0.005 of sulphuric acid, gives a very evident yellowish colour. Citric acid, whether concentrated or diluted, gives a fine yellow colour, which is as durable as that occasioned by phosphoric acid. Tartaric acid also gives a very fine yellow colour, but it soon fades and becomes dull, in proportion to the weakness of the acid. If, for instance, it is diluted with five parts of water, it gives a less lively colour than citric acid mixed with 15 or 20 parts of water. Malic acid acts nearly like tartaric acid. Concentrated oxalic acid produces an orange colour which becomes gradually yellow. Diluted with one part of water, it gives a yellow colour, which remains pretty good. If the acid is diluted with three parts and more of water, the yellow colour at first produced disappears in a few minutes. Succinic acid gives a yellowish colour which soon fades, and benzoic acid has scarcely any action. It occurred to me that the fine yellow colour which succeeds the red colour of Brazil wood, when it is subjected to the action of phosphoric or citric acid, might be employed in the art of dyeing. To ascertain this, I tried at several times to dye wool by means of the above-mentioned substances, and these trials afforded results which exceeded my expectation. Some woollen manufacture dipped into a boiling bath of Brazil wood, acquired a yellowish-red colour, but it was dull. If, after washing and draining, it is dipped for a few minutes in a boiling and very dilute solution of phosphoric acid, or lemon juice, diluted with water, a very bright yellow colour is immediately produced. As phosphoric acid is too dear a substance to be employed with advantage in dyeing, I tried as a substitute acidulous phosphate of lime obtained by treating bones with sulphuric acid, and I found that this substance acted precisely in the same manner, and gave as fine a colour as that produced by pure phosphoric acid. Woollen manufacture dyed yellow, by means either of an acidulous phosphate or lemon juice, may be subjected to the strongest soaping, without the colour undergoing any alteration. I have had no opportunity of ascertaining by direct experiment whether this colour is permanent, and resists the action of the sun; it may be admitted that even if the colour produced by the action of lemon juice is not permanent, that given by the acidulous phosphate, on the contrary, will be so, as being a combination of the colouring principle with a substance which is perfectly unalterable by water, air, or heat. Silk is also susceptible of receiving a fine yellow colour by the process which has been described; but as to cotton and linen, the very incomplete experiments which I have had an opportunity of performing, have not afforded a satisfactory result; it might, perhaps, be possible to succeed if the substance to be dyed was previously animalized. It is, however, worthy of remark, that paper, as I have already observed, receives and retains this colour with all its brightness. ARTICLE VIII. Astronomical Observations, 1822. Bushey Heath, near Stanmore. Latitude 51° 37′ 44.3" North. Longitude West in time 1' 20.93". N. B. The observation of the 24th uncertain to five seconds. ARTICLE IX. An Investigation of the Method for finding the Sum of all the Coefficients in the Expansion of a Multinomial. By Mr. S. Jones. (To the Editor of the Annals of Philosophy.) SIR, Nash Grove, Liverpool, May 29, 1822. It is a remarkable coincidence that two of our most general theorems in analytical calculations should have been published by their respective authors, Dr. Brook Taylor, and Sir Isaac Newton, without their demonstrations: the utility of the former in the differential and integral calculus, vanishing fractions, the higher mechanics, &c. has induced many of the continental, as well as several of our own mathematicians, to attempt to demonstrate it from first principles; it is, however, generally acknowledged that its difficulty excludes that conciseness, perspicuity, and native simplicity, which all fundamental propositions ought to possess; the latter, or binomial theorem, though less difficult, was, at the time of its discovery, no easy task to demonstrate; accordingly we find that it caught the attention of the most eminent mathematicians, and has employed the talents of Maclaurin, Simpson, Demoivre, Euler, Lagrange, Woodhouse, and others, to whose minute researches and amplitude of remark it might be supposed that nothing more could be added; yet it is presumed that to demonstrate generally what only one of these, Euler has done for a particular case, and that by numeral induction, will not be thought an inelegant appendage to this beautiful theorem, nor unworthy the attention of the mathematicians of the present day. For the sake of simplicity, I shall divide the proposition into the following cases, beginning with the easiest. 1. To find the sum of all the coefficients in the expansion of a monomial a, to the power of n. It is manifest that a" = a xa xa x to n terms, and since a has unity for its coefficient, the sum will be 1x1x1x1 to n terms = 1"; therefore, the sum of all the coefficients in the expansion of a monomial is 1". 2. To find the sum of all the coefficients in the expansion of a binomial a + b, to the power of n. By the binomial theorem, (a + b)" = a" + n . a"−1 b + n . "— 1‚ a"-2 b2 + n . n-l n 2 n-2 a"-3 unity for its coefficient, all the powers of b will have unity for their coefficients, which consequently will not affect their sum; whence they may be rejected, and the sum of all the coefficients in the expansion of (a + b)" will be a" + na" ~ ' + n. N 2 12-12-24 a"-3+ &c.; but a being a monomial, 3 the sum of all the coefficients in a" is 1", a" is 1-, a"-2 is 1-2, a"-3 is 1-3, &c. These values of a", a"-', a"-2, &c. being substituted in the expansion, gives 1* + n . 1*~1 + n .”- 1 . 1*~2 + n . 2 1-3 + &c. but this series is absolutely the expansion of (1 + 1)" 2"; therefore the sum of all the coefficients in the expansion of a binomial a + b, to the power of n, is 2". 3. To find the sum of all the coefficients in the expansion of a trinomial a + b + c to the power of n. To obtain this, put a + b = x; then (a + b + c)" = (x + c)” + &c.; and because c has unity for its coefficient, all its powers may be rejected, and the sum of all the coefficients in the expan sion of a trinomial will be x" + n.x” &c.; but since r is a binomial, the sum of all the coefficients in "= (a+b)" is 2", x"−1 = (a + b)"-1 is 2"-1, 2 x"2 = (a + b)-2 is 2"-2, &c. These different powers of 2 being substituted for x"; x2-1, |