mingled with an alkali and then with an acid. The same treatment does not form any prussic acid. In conclusion, I take the opportunity of recording a few observations which I have made on the action of iodine, of hydriodic acid, and of sulphuretted hydrogen, on prussic acid, on cyanuret of mercury, and on sulphuretted chyazate of copper. Iodine decomposes the aqueous solution of prussic acid, and becomes hydriodic acid, cyanogen being at the same time evolved. On the contrary, hydriodic acid is itself decomposed by cyanuret of mercury, red ioduret of mercury and prussic acid being formed. The affinity of mercury for iodine doubtless determines this decomposition. Iodine, when put into a solution of cyanuret of mercury, sets the cyanogen at liberty, and forms red ioduret with the metal. Sulphuretted hydrogen gas, when quite dry, does not appear to act on sulphuretted chyazate of copper; but it instantly decomposes it when water is present, sulphuretted chyazic acid being separated, and sulphuret of copper formed. Tower, April 3, 1819. R. PORRETT, JUN. ARTICLE V. New Demonstrations of the Binomial Theorem. By Mr. Herapath. (To Dr. Thomson.) AMONG the many demonstrations that have been given of the binomial theorem, I do not remember to have seen one that is both elementary and complete. That in the Calcul des Fonctions is, perhaps, one of the most elegant and complete that has yet been given; but it has been objected to as not being elementary. The same objection might, with a little modification, be made to one or two neat demonstrations that have appeared in some of the late volumes of the Philosophical Transactions, and to others that I have met with in different authors. It seems that mathematicians have considered the lower branches of algebra to be quite insufficient, without some assistance from the higher analysis, to effect a proof of this celebrated theorem. Whether Newton's not attempting to demonstrate this, one of the most beautiful and valuable of his mathematical discoveries, and his resting satisfied of its general truth merely from trials in a few particular cases, may have had any influence, I will not take upon me to determine; but I hope the following demonstration, drawn from common algebra, will show, that there is no necessity of having recourse to other principles to obtain a 2 365 proof simple, direct, and complete. If it has any other merit, it is to the best of my knowledge that of novelty. Knowle-hill House. J. HERAPATH. Of the Invention of a Theorem for raising a Binomial to any whole positive Power. Let a + b be any binomial, then if we take the successive powers of it by actual multiplication, there will result, 5. a5+5 a4 b+ 10 a3 b2 + 10 a2 b3 + 5 a b4+ 6. a6+6 a5 b + 15 a1 ba÷20 a3 b3 + 15 a2 b4 +6 a bɔ+ b6 1, 2, 1. 1, 3, 3, 1. 1, 4, 6, 4, 1. 1, 5, 10, 10, 5, 1. from which it is manifest, as well as from a consideration of the process of multiplication by which those coefficients are produced, that the coefficient of any term is equal to the sum of the coefficients of the corresponding and preceding terms of the next lower power; as, for instance, in the sixth power, 1 = 1 +0, 6 5 + 1, 15 = 10+5, &c. Therefore the coefficient of the second term of any whole positive power n, must be n; that of the first term being unity. Again, if we divide the coefficients of the third terms by those of the second, we shall have Coefficients of second terms or exponents of the powers... Quotients of third by second coefficients.. n-1 } 1, 2, 3, 4, 5, 6, 7.. n-1 2 2 3 4 5 bbb bb. Whence is the general multiplicator by which the third co 2 efficient is produced from the second. Consequently the power being n, the coefficient of the third term is n × x n-1 And generally, by following the same method of dividing the succeeding by the preceding coefficients, we shall have Indices or coeff. of 2d terms 1, 2, 3, 4, 5, 6, 7, 8, 9.. n or Therefore it is evident that the multiplicators by which the succeeding coefficients are generated out of the preceding are the terms of this series", "-1, 1-2, 1-3 .... n-g; and consequently the general theorem for raising a binomial to any whole positive power is a" + na"-1b + n."-1 n-1 a2 - 1 b2 + n any thing of the law of the exponents of a, b, because it is too obvious to require elucidation. In this way would the invention of the theorem have been much more natural and easy, at least for whole positive powers, than the method of interpolations followed by Newton; and it being obtained for this case, the others are easily deduced from it. Thus for the case of bs.... &c. which shows that the general theorem for positive exponents, being whole numbers, is equally true for negative. The same principle discovers the form of the theorem when the exponent of the power is fractional. For if the terms of the fraction be m, n, and we have x + y = (a + b)"; by involution we get (x + y)" = (a + b)"; and + (1) And since (x + y)" = (a + b)", if we expand each side and equate the first terms, there will result x = a (2) Now since x + y = (a + b)", it is evident that the terms of equation 1, properly taken, ought to give the development of (a + b)"; and this will be the case, if we collect the terms in the vertical order of their collocation, and make the necessary substitutions from equations 2 and 3. The first term gives =a"; and the second term, without its coefficient, will be a" b; and thus it may easily be seen that the powers with a fractional positive exponent follow the same law as those whose exponents are whole numbers. Therefore, for the sake of brevity, we shall omit the powers of a and b, and consider the coefficients only. By substituting from equation 3, the coefficient of the second In like manner the coefficient of the third term, by taking the coefficients of the three next vertical terms, is equal to n But here we have taken no notice of the coefficient of the second term m-1 x bs n - 2 -1 m2 x ba of the value of y, which was neg lected in the former substitution for the coefficient of the second term. If, therefore, we add this coefficient, multiplied by -n- -1 the coefficient of in equation 1, to."" found from sum ming the three vertical terms, we shall have n 2 m m-n n 2 n for the en tire coefficient of the third term of the development of (a+b)^; that is, the same as would be given by the general theorem for whole positive powers by changing n into. And by following the same course we shall discover a like coincidence in the coefficients of the fourth and other terms. Whence the theorem is true generally for positive fractions; and that it is equally so for negative may be shown by common division; therefore, it is universally true for all whole and simple fractional numbers. Because all fractions, whether mixed, compound, or continued, -may be reduced to simple fractions, having their numerators and denominators whole numbers, and the theorem has been proved to be universally true for such fractions; and because every irrational number may be either accurately or so nearly expressed by a fraction that the difference shall be less than any assignable quantity, it follows that the theorem is true for all numbers rational or irrational. To extend this theorem to imaginary exponents, we must observe, that as the form of an irrational exponent is not changed by making it imaginary, so neither is the form of any coefficient which is a function of this exponent; consequently the theorem is likewise true for imaginary powers, and is, therefore, universally true. A demonstration of the binomial theorem might easily have been given for fractional powers, by pursuing the same route that I have for the demonstration of whole numbers; namely, by extracting the successive roots, and observing the law which connects the quotients of the coefficients of the succeeding by those of the preceding terms. I have, however, chosen the present method, because it is more simple and natural, and because it exhibits a connective dependance between the proofs of whole and fractional numbers that was supposed not to exist, and displays the resources of the elementary branches of a science, which has itself, for this purpose in its full extent, often been thought to be not sufficiently general. ARTICLE VI. ANALYSES OF BOOKS. Recherches sur l'Identité des Forces Chemiques et Electriques. Par M. H. C. Ersted, Professeur a l'Université Royale de Copenhagne, et Membre de la Societé Royale des Sciences de la même Ville, &c. Traduit de l'Allemand par M. Marcel de Serres, Ex-Inspecteur des Arts et Manufactures, et Professeur de la Faculté des Sciences à l'Université Imperiale; de la Societé Philomatique de Paris, &c. Paris, 1813. In the fifth volume of the Annals of Philosophy, p. 5 (Jan. 1815), I gave some account of this work, mentioning at the same time that I had not seen the book itself, but derived my information from the German journals, and from an outline given by Von Mons in his translation of Sir H. Davy's elementary work on chemistry. Some time after this notice of mine appeared, I received a letter from Professor Ersted informing me that the account of his book in the German journals was far from accu |