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Nor is it only in the recesses of this abstract science that its advantages will be felt : it is peculiarly adapted to the discovery of those laws of action by which one particle of matter attracts or repels another of the same or of a different species; consequently, it may be applied to every branch of natural philosophy, where the object is to discover, by calculation from the results of experiment, the laws which regulate the action of the ultimate particles of bodies. To the accomplishment of these desirable purposes it must be confessed that it is in its present state unequal; but should the labours of future enquirers give to it that perfection, which other methods of investigation have attained, it is not too much to hope, that its maturer age shall unveil the hidden laws which govern the phénomena of magnetic, electric, or even of chemical action.'

Without entertaining ideas so sanguine as those of Mr. Babbage on this subject, we can perceive that an immense advantage would be gained, and an almost indefinite extent given to the analytical sciences, if the calculus to which he here alludes could be brought to perfection: but, unfortunately, we see little prospect of this being effected. Mr. B.'s two memoirs occupy a considerable number of pages, and we must say that few advances are made towards the completion of his views. We think that, by this time, he must be well aware of the great difference between forming a project and carrying it into execution; and he must have discovered that it is one thing to be convinced of the utility of a calculus, and another to reduce it to a manageable shape, to fix its limits, and to investigate and establish its rules. : In our remarks on this gentleman's former paper, we expressed some apprehension of a complicated notation and an artificial solution; and we conceive that no person, who looks even slightly at the pages of the present article, will say that our apprehensions were groundless. Every case seems to require distinct symbols; and new characters start up in nearly every problem, with which the mind becomes bewildered, embarrassed, and fatigued. If any tolerably simple means could have been devised for the solution of problems of the kind proposed by Mr. B., it would, as we before observed, have been an important improvement in analysis; and we were willing to entertain hopes when we reporteri his last paper, that in the present article, which was then promised, we should have seen a direct application of his calculus to the solution of some simple and obvious examples: but we do not think that those which are given can be considered as falling under this class. Of the problems, the second is perhaps the simplest, as well as the nearest in character to those in the former mem oir; and it may therefore be understood by those

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arrive at this conclusion if the case had not been exhibited ? It is true that we might by chance have stumbled on the solution : but the matter required is a direct process, which Mr. Babbage does not appear to have acquired.

A New Demonstration of the binomial Theorem. By Thomas Knight, Esq. — The author begins this memoir by observing that

It is somewhat remarkable that, amongst the various and far-fetched methods and artifices by which the binomial theorem has been obtained, no one should once have thought of the only course which seems obvious and natural. The equation (a +x)m * (a + y)" = {(a + x) (a + y)}" expresses the general property of powers, whether m be positive or negative, whole or fractional, and from this equation, without the help of any artifice, the series in question is deduced.

. Some investigations have been found fault with, as drawn from principles allied to the method of fluxions; whilst, on the other hand, a demonstration, taken from the “ Théorie des Fonctions," has been represented as perfect; but I cannot help thinking that it is as much connected with the fuxional calculus as any of the rest; for it seems to make no difference whether, 'in (a + x)" we substitute x+u for x, and take the co-efficient of u, or substitute x + x, and take the co-efficient of ċ. The former substitution was made because it was known to be equivalent to the other, and has so little apparent connection with the subject, that a student would hardly understand why it was made. The demonstration of M. La Croix in the Introduction to his " Calcul Différentiel,” is liable of course to the same objection. If we multiply a + x, continually, first by itself, and then by the powers successively arising, we easily see that the second term of each succeeding product is of the form na "-*x, n being the exponent of the power: this does not require a more formal proof, and I assume it in what follows. Nor is it more difficult to perceive, that, generally m being positive or negative, whole or fractional, the following form may be assumed.

(a + x) = a + Aam-5x + "Aam-*2 + "Aam-3x} + where 'A, "A, ""A, &c. are expressions depending on m alone.' In the same manner,

(a + y)" = a" + 'Aam-* y + &c. and therefore, by comparing (a + x)" X (a + y)m, with (a2 + ax + ay + xy)" or with am (a + x + ay), by first making *=1+, and retaining only the first power of y in both expansions, the forms or values of the co-efficients 'A, "Ą, &c. become determined.

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