Tried with the Geranium, Myrtle, and Polypodium, so far as regards its relative influence in causing the emission of oxygen.

II. Artificial light, obtained A. from lamps. Tried by Professor DECANDOLLE.

B. from incandescent lime.

influence detected.

Tried by myself, but no

PART II.-ON THE ACTION OF PLANTS UPON THE ATMOSPHERE,

II. Proportion between the carbonic acid absorbed and oxygen evolved.

III. Greatest amount of oxygen

My experiments show that when plants are confined the former is always greatest at first; but this may not continue to be the case after a certain interval.

that can be added to the air of My experiments show that at least 18 per cent. of a jar by the influence of a plant.

IV. At what stage in the scale of vegetable life the function of purifying air stops.

oxygen may be so added.

Probably where there cease to be leaves. I have shown that it exists in dicotyledonous and monocotyledonous, in evergreens and deciduous, in terrestrial and aquatic plants, in the green parts of succulents, as well as in ordinary leaves, in Algæ and in Ferns as well as in phanerogamous families, Prof. MARCET has shown that it does not take place in Fungi.

XIV. Researches on the Integral Calculus, Part I. By H. F. Talbot, Esq., F.R.S.

On the ulterior object of these researches, we cannot pretend to offer a conjecture; but so far as they at present extend, they are full of interest. We are not, however, without hope that a course of new inquiries is here opened, which will in the end lead to the solutions of various physico-mathematical problems which have resisted the united efforts of all geometers who have hitherto attempted them.

It is familiarly known that every problem which natural philosophy presents to the mathematician is ultimately, and without much difficulty, reduced either to the solution of an equation, or to a series of integrations. So far as "algebraic equations" are concerned, this difficulty has been entirely removed by living geometers, Horner and Sturm. In the case of transcendental equations, no general method of direct solution is known; but the rule of Double Position is found in most cases to be effective, though the immense labour which it requires is often too great to admit of its application to the cases which present themselves. Much, therefore, remains to be done; and most probably the ultimate method of solution of such problems that will be arrived at is, the invention of methods of developement in rapidly converging series, a few of whose terms may be taken as a sufficient approximation to the sum of the whole.

The differentiation of a function can, in general, be easily effectedas is the case with all direct mathematical processes; but the differential expression itself that is given, whether taken at hazard, or resulting from the expression, in mathematical symbols, of a physical hypothesis, may not be such as could have resulted from any direct differentiation. In such case, we may infer at once that its integral cannot be exhibited. Such an expression can, however, always be integrated if conjoined with some other expression of a more or less complicated character. But this does not alter the state of the difficulty, except the expression thus added to the given one does itself admit either of immediate integration or of a similar reduction into an integrable part and another which is similarly reducible, so that by continuing this process we at last arrive at an expression which is wholly integrable.

Of course it is difficult to say, beforehand, what expressions are integrable, since an affirmation of that nature would imply a knowledge of the differentials of every possible expression to be present to the mind, a supposition that would be perfectly absurd. There is, indeed, a test, whether an expression of several variables be an exact integral or not; but this does not apply to the case of a single variable; and even in the cases to which it does apply, its application is sometimes very difficult, from the different forms under which its results are exhibited. What is still worse, when the integrating factor is found, we have still scarcely any clue furnished by it to the ultimate solution of the differential equation.

As to the method of integration by series, that is, by developing the function attached to do in series, though by different artifices rendered available to a great number of actual problems, it is not in a

general condition to be adapted to the wants of the inquirer, inasmuch as the instances are very few in which the series is sufficiently convergent to be calculated in numbers; and in no case does it furnish any material assistance in judging of the general character of the phenomenon it is intended to express, or of the analytical nature of the function itself. A great number of physical problems, viewed merely as such, are sufficiently solved if we can ascertain the value of the integral, when taken between certain limiting values of the variable. This has given rise to the calculus of definite integrals. This method, however, as a mathematical process, is still in its infancy; though some specimens, given by Mr. Murphy in the Cambridge Transactions, of attempts at a general method, lead us to hope that it may yet be rendered more perfect and effective than it at first seemed capable of becoming.

The inquiry respecting the algebraic expression of the values of integrals between given limits-or rather requiring the second limit to be found, the first being given-originated with Fagnani, about 1714. He published, in 1715, in the Giornale de Letterati d'Italia, a solution of the problem-given the equation of the parabolic curve "y (where n is 3, 4, 3, †, §, or 4) and an arc of it, to find another arc of it, so that their difference may be rectifiable." Again, in 1718, he published a variety of important theorems respecting the ellipse, hyperbola, and lemniscate, in which he showed how to find two arcs whose difference should be a straight line. This gave rise to a more extended inquiry.

dx

does not admit of an

The expression √ a +ßx + yx2 + dx3 + €x* exact integral directly obtainable by any known means; but Euler showed that by taking another function in y similar to this in its form, and identical in its constants, the sum or difference of the two might be integrated in an algebraic form. Many important consequences followed from this result, and many attempts to extend the method to other forms of the radical, and to a greater number of terms, have been made by subsequent mathematicians. In 1792, Legendre read to the French Académie a memoir* whose object was to classify and arrange the elliptic integrals which were implicitly contained in Euler's solution. He subsequently expanded this paper, and published these extended researches in a work entitled Exercices du Calcul Intégral, in three vols., quarto, Paris, 1811. His tables are peculiarly valuable.

(n being any whole number), +a, xm. the lamented 1828, gave an expression for the sum of

Putting R = x2 + a x2−1 + and P a function of the same form, x' Abel of Christiana, published in a series of integrals of the form

Pdx

d

✓ R

arrived at his formula were never published.

The steps by which he

In 1834, Poisson, in a paper published in Crell's Berlin Journal, has considered, several forms of such integrals, which are not comprehended

• In 1809, a translation of this valu- |logy of the differential calculus into English able memoir was printed in Leybourn's books. This was at least ten years prior Mathematical Repository; and it is more- to its introduction into the Cambridge ever remarkable as containing the first works. introduction of the notation and termino

in Abel's theorem; and it hence appears that this celebrated result is not given in all the generality of which it is susceptible.

This is the state in which Mr. Talbot found the problem;—that is, when he came to examine what had been done by others, though he had obtained his chief results, and was in possession of his general method several years before Abel's theorem was made public. The problem which he here proposes to solve is:-"To find the sum of a series of such integrals as ƒ (R)dx, R being any entire polynominal, and ☀ any function whatever."

66

The history illustrated by examples which he gives of his own progress in these researches, is highly judicious and instructive. It is, however, impossible to give a condensed and intelligible account of his processes within the limits of this Magazine: nor, till the whole series is before us, would it be possible to do the method that full justice which it very obviously demands. Two methods of proceeding have already been developed,-the one of which is founded on a change of the conditions," shown to be necessary in the other for the solution of the problem in all its generality. The whole process is founded on the method of "integrating by parts," a series of symmetrical functions of the assumed variables. Yet we would not have our readers think, because its first principles are known ones, that its results have ever been anticipated, or its current processes ever employed before.

This paper terminates the part; and we would earnestly recommend its attentive perusal to every mathematician who feels interested in the

progress of his science. We may state, moreover, that it is not a difficult paper to read. There is none of the quackery of new symbology, or of the mystified and unsatisfactory reasonings which so much disgrace too many of "the most learned" mathematical papers of the present day, to be found in this; and no acquaintance with other writers, beyond the mere elementary ones, is required to enable the reader to comprehend it fully.

ON THE SIGNS OF MULTIPLICATION,
DIVISION, ETC.

In our fifth number, (vol. i., p. 291,) we offered a conjecture respecting the origin of the signs + and Since then, the writer has received from a friend, the most eminent mathematical antiquary of the present age, and a distinguished professor in the University of Oxford, another conjectural mode of deriving these symbols. An extract from his letter is placed below; and we proceed to lay before our readers the best information that is possessed respecting a few others.

The Symbol of Multiplication, x, was first used by Oughtred; and he prefaces its introduction simply with the remark :-" Multiplicatio speciosa connectit utramque magnitudinem propositam cum notâ in vel x; vel plerumque absque nota, si magnitudinis unica litera. Et si signa sint similia, producta magnitudo erit affirmata: sin diversa negata.

Effertur autem per in.*" It probably was used as a variation of the form of the symbol +, the operation of multiplying being merely a substitution for that of addition, in the case where all the numbers to be added are equal to one another. We had, indeed, before we were aware of this passage, imagined it to be a contracted representation of the "hand-in-hand," and to designate perfect union or amalgamation. That passage, however, does not seem compatible with such an hypothesis; as in such case, some remark on the subject would in all probability have been made to point out the views by which he was led to it, and to enforce its adoption by others.

The Symbol of Division,, is merely a contracted mode of designating the positions of the divisor and dividend in the old Italian mode of operating. It was employed in the first place to concentrate the matter on a printed page; as it requires two lines to express it fractionally, and only one to express it by means of this interposed symbolt.

The Sign of Equality, in its present form, =, was first employed by Recorde. He gives his reason thus:-" and to avoid the tediouse repetition of these woordes:` is equalle to: I will sette as I often doe in woorke use, a paire of paralleles or gemowe lines of one lengthe, thus, =, because noe two thynges can be moare equalle."—Whettestone of Witte, p. 105. Harriot, also, apparently used it, without any knowledge of Recorde having done so before him.-Ars Praxis Analytica, p. 10.

Before this time, and for a long period subsequently on the continent, the symbol of equality, was °C, or 0, which is very evidently the initial diphthong æ, of æqualis ‡.

The Symbols of Greater and Less, viz. > and <, were invented by Harriot, and first appeared in Warner's publication of the Ars Praxis Analytica, some years after the death of that extraordinary man§. See that work, p. 10. They are very appropriate, the point being in both cases directed towards the less quantity, and the opening towards the greater. The sign of inequality, without asigning which is the greater, viz.,, is of modern date, and we are not quite certain who was the first to use it. It is merely the sign of equality," crossed out." Girard used ff and § for greater and less, or for > and <.

As a conjecture respecting the symbol placed between two quantities to signify their difference without assigning which was the greater, we have heard a very eminent scientific gentleman express his opinion that it is the letter s, employed as the initial of subtrahere: but we rather incline to think it a modification of the manuscript d. Of this the reader may easily satisfy himself by writing the small d with the