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in 'a mere combustion at the surface of contact of the inflammable matter,' but rather as the combustion of an explosive mixture of inflammable gas, or vapour and air.' On this principle we are to explain the extraordinary results of Sir H. Davy's experiments. A considerable bulk of heated metal is necessary to raise the temperature sufficiently high to inflame an explosive gas; so that, by passing through the wiregauze, even when red hot, the flame parts with so much of its heat that it is no longer hot enough to produce the explosion. Sir Humphry had before shewn that the explosive mixture required a very high temperature for its combustion; and there is reason to believe that the heat of flame is as great • as any with which we are acquainted,' - in course much greater than that of hot iron.

We cannot leave these papers without expressing the high gratification that we experience, when we observe the immediate practical benefits of philosophical discoveries ; and few cases are on record, in which an evil of such magnitude was so effectually obviated, as the destructive explosions of the coal-mines promise to be through the use of the lamps of Sir H. Davy.

Some Observations and Experiments made on the Torpedo of the Cape of Good Hope, in the Year 1812. By J. T. Todd, late Surgeon of his Majesty's ship Lion.- Mr. Todd informs us that the torpedo is frequently caught in Table Bay, to the westward of the Cape, but very seldom in the bays to the eastward. The columns of the electrical organs, in the fish which he examined, were larger and less numerous than those that were described by Hunter at Rochelle. He sums up the result of his observations in a series of propositions, which we shall quote at full length; since they contain all the information that is dispersed through the paper, and, for the most part, may be regarded as being correct deductions from the premises.

* The preceding account appears to me to afford grounds for the following conclusions.

* 1. That the electrical discharge of this animal is in every respect a vital action, being dependent on the life of the animal, and having a relation to the degree of life and to the degree of perfection of structure of the electrical organs.

• 2. That the action of the electrical organs is perfectly voluntary.

That frequent action of the electrical organs is injurious to the life of the animal ; and, if continued, deprives the animal of it. Is this only an instance of a law common to all animals, that by long continued voluntary action they are deprived of life? Whence is the cause of the rapidity with which it takes place in this instance? Or is it owing to the re-action of the shock on the animal?

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4. That those animals, in which the nerves of the electrical organs are intersected, lose the power of communicating the shock, but appear more vivacious and live longer than those in which this change has not been produced, and in which this power is exerted. Is the loss of the power of communicating the shock to be attributed to the loss of voluntary power over the organ? Does this fact bear any analogy to the effects produced by castration in animals ?

5. That the possession of one organ only is sufficient to produce the shock.

' 6. That the perfect state of all the nerves of the electrical organs is not necessary to produce the shock.

' And, 7. From the whole it may be concluded, that a more intimate relation exists between the nervous system and electrical organs of the torpedo, both as to structure and functions, than between the same and any organs of any animal with which we are acquainted. And this is particularly shown, ist, By the large proportion of nerves applied to the electrical organs: and, 2dly, By the relation of the action of the electrical organs to the life of the animal, and vice versa.'

Some Account of the Feet of those Animals whose progressive Motion can be carried on in Opposition to Gravity, By Sir Ev. Home, Bart. V.P.R.S. --- Every one must have observed the power which the common bouse-fly possesses of walking on the ceiling of rooms, although few persons have thought of inquiring by what means it is enabled to support itself in opposition to gravity, and at the same time so readily to exercise progressive motion. The minuteness of the fly's foot, indeed, renders it difficult to determine this point: but a particular kind of lizard, the Lacerta Gecko, a native of Java, has the same power, and a foot so large as to be easily examined. The foot of that animal is accordingly described by the present author; and we learn that it consists of five toes, each toe being furnished with sixteen transverse slits, communicating with as many cavitics : as also that it is provided with a muscular apparatus, by the contraction of which the cavities are opened, so that the animal rests on the serrated edge with which these cavities are surrounded. Sir Everard remarks that the apparatus of this lizard's foot bears a considerable resemblance to that part of the head of the Echineis Remora, or sucking fish, by which it attaches itself to ships or other solid bodies; and that we may conclude that they both act on the same principle. The mode of action is thus described:

• It is evident, that when the external edge of this apparatus is closely applied to any surface, and the cartilaginous plates are raised up, the interstices must become so many vacua, and the serrated edge of each plate will keep a sufficient hold of the substance on which it rests, to retain it in that position, assisted by the pressure of the surrounding water, without a continuance of muscular exertion.

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• It thus appears, that the adhesion of the Echineis Remora is produced by so many vacua being formed by an apparatus worked by the voluntary muscles of the animal, and the pressure of the surrounding water.'

Though the minuteness of the fly's foot renders it very difficult to determine precisely its mechanical structure, the author deems it highly probable that certain concave surfaces, which have been observed attached to it, are employed to form vacua,

which enable the animal to move under such disadvantageous circumstances, upon the same principle as the Lacerta Gecko.'

MATHEMATICS, ASTRONOMY, and OPTICS. On the Developement of exponential Functions ; together with several neue Theorems relating to finite Differences. By John F.W. Herschel, Esq. F.R.S.--- We have frequently had occasion to observe that the analytical sciences in this country were rather retrograding than advancing, and that nothing strikingly new and interesting had for many years issued from the English press on those subjects; while the transactions of foreign academies, and particularly those of the French Institute, have abounded with valuable and brilliant discoveries. These, however, have not been unmixed with matters of mere curiosity and difficulty; the purpose of the writers, in many cases, being obviously to shew their own dexterity in the transformation of quantities and equations, and to make a great display of intricate and almost unintelligible formulæ, without the least consideration of their application to any purpose of real utility. If, therefore, any desire should arise, as we think we can now perceive that it does, among our English mathematicians, to emulate the same class of men in France, it will be of the highest importance to embrace only such subjects as will admit of useful application ; and to bear in mind that it is not the intricacy of formulæ, but the simplicity of them, which constitutes the beauty of analysis.

Mr. Herschel has in two or three instances manifested considerable analytical talents, which we should be very loth to undervalue: but we fear that he is too fond of that sort of parade to which we have alluded, and which we should be glad to see him correct. We wish it also to be understood that these remarks are not so much intended to apply exclusively to the present article, as to the general character of his recent communications to the Royal Society, and to a Cambridge work in which he is supposed to take an active part.

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The subject of the paper before us is the developement of exponential functions, the origin and progress of which are thus stated in the introduction to it:

• In the year 1772, Lagrange, in a memoir published among those of the Berlin Academy, announced thuse celebrated theorems expressing the connection between simple exponential indices, and those of differentiation and integration. The demonstration of those theorems, although it escaped their illustrious discoverer, has been since accomplished by many analysts, and in a great variety of ways. Laplace set the first example in two memoirs presented to the Academy of Sciences *, and may be supposed in the course of these researches to have caught the first hint of the Calcul des Fonctions Generatrices with which they are so intimately connected; as, after an interval of two years, another demonstration of them, drawn solely from the principles of that calculus, appeared, together with the calculus itself, in the memoirs of the Academy. This demonstration, involving, however, the passage from finite to infinite, is therefore (although preferable perhaps in a systematic arrangement, where all is made to flow from one fundamental principle) less elegant; not on account of any confusion of ideas, or want of evidence; but, because the ideas of finite and infinite, as such, are extraneous to symbolic language, and, if we would avoid their use, much circumlocution as well as very unwieldy formulæ must be introduced. Arbogast also, in his work on derivations, has given two most ingenious demonstrations of them, and added greatly to their generality; and lastly, Dr. Brinkley has made them the subject of a paper in the Transactions of this Society, to which I shall have occasion again to refer. Considered as insulated truths, unconnected with any other considerable branch of analysis, the method employed by the latter author seems the most simple and elegant which could have been devised. It has however the great inconvenience of not making us acquainted with the bearings and dependencies of these important theorems, which, in this instance, as in many others, are far more valuable than the mere formulæ.

• The theorems above referred to are comprehended in the equation Ax.D

(a) or, more generally,

Δ.r.D .

(6) where the A applies to the variation of x, and the D to the functional characteristic u; and where n may have

any value whatever.' In this form, these theorems are obviously no more than abridged expressions of their meaning; and therefore, in order

« * Mém. des Savans Etrangers, 1773, p. 535. - Mém. de l'Acad. 1772, p. 102.'

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to become practically useful, their second members must be developed in a series of the powers of A x. D.- Mr. Herschel

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f this = A + At + + &c., he arrives, after several transformations, at the following theorem, viz. f(+4)=f(1)+ 6 (1+)0+

oth $(1+4)0*+&e. in the application of which to any particular case, it is necessary to develope f(i + A) in powers of A; then, striking out the first term, as well as all those in which the exponent of A is higher than that of t, to apply each of the remaining terms immediately before the annexed power of o; and the developement is then in á form adapted to numerical computation.

The author next proceeds to shew the application of the above and other equivalent formulæ to the actual developement of some exponential functions, which he is enabled to perform with great facility : but neither the nature nor the limits of our work will allow us to follow him farther in his transformations.

On new Properties of Heat, as exhibited in its Propagation along Plates of Glass. By David Brewster, LL.D. F.R.S., &c. The science of physical optics has within a few years assumed a new and highly interesting form, and no one has contributed more to stamp it with its present importance than the ingenious author of this paper: nor has he, in any of his preceding experiments and researches, developed a series of more striking and curious phænomena, than those which are here presented for the contemplation of the philosopher.

In a former number, we reported Dr. Brewster's experiments and deductions relative to the action of heat in enabling glass to arrange a beam of light into two opposite polarized pencils; in which he has shewn that "unannealed glass, in the form of Prince Rupert's drops, possesses distinct optical axes, and acts on light like all regular crystallized bodies.' It appears that his attention was again called to this subject, in consequence of his having discovered that reflection from all the metals, and total reflection from the second surface of transparent bodies, produced the same effect as crystallized plates, in separating a beam of polarized light into its complementary tints.

This circumstance led the Doctor to suppose that the existence of the two opposite polarized pencils, and the production of the complementary colours, were concomitant effects: he was in consequence induced to examine the

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