condition is partly produced and maintained by the various secretions that take place in the animal body. How far the fluidity of the blood, and the vitality of the blood, as it is called, are dependent upon this electric state or condition, are questions which must necessarily arise in our minds. The particles of the blood, also, must under these circumstances exist in a state of self-repulsion; and may not this fact, it may be asked, tend to explain some of the phenomena connected with the circulation of the blood in parts not dependent upon the vis à tergo action of the heart, and also those connected with the coagulation of the blood when taken from the living animal? These are questions that will arise; but I must not wander too far from our present object; and therefore conclude this Section by stating that a clue has now been obtained to the non-appearance of any effect upon the galvanometer when the two electrodes are inserted into an artery and a vein, a fact previously established by the experiments of POUILLET and MULLER*. As the blood in the two vessels is in the same electric state, no effect could occur upon the needle; thus proving the fact, well established by FARADAY, that in order to obtain CURRENT FORCE the circuit form must be given to the arrangement, i.e. that the electrodes must be brought into contact, or by means of some conducting mass, with the ANION and CATION originating the powert.

Before entering upon the concluding remarks there are one or two points which must be noticed. It may be supposed, 1st, that the effects that have been obtained may arise from thermo-electric actions, since BECQUEREL and BRESCHET have ascertained the existence of a difference in temperature between the arterial and venous blood by means of a galvanometer; 2ndly, that they may also arise from the actions that take place upon the surface of the platinum electrodes. There can be no doubt that a part of the effects may be referred to both of these circumstances, and they must therefore be taken into consideration when judging of the final result upon the needle. As these objections have however been already noticed in one of the original papers §, I cannot do better than refer to the experiments and arguments there brought forward for their refutation.

Concluding Remarks.

The results recorded in the present and previous papers tend to establish the following conclusion, viz. that the act of secretion in the living animal is accompanied with the manifestation of CURRENT FORCE; and the phenomena with which this act of secretion appears to be the most intimately related are those that occur in the voltaic circle, as I have endeavoured to point out in the present paper. A difficulty may arise to some minds in perceiving this relation, from the circumstance that in the ordinary voltaic circle metals are employed. If we bear in mind that the metals, although one of them is usually acted upon, serve principally as conductors, and that they are not essential for the development of the power, this difficulty will be easily removed. Now as the manifestation of current force during the actions which occur in the voltaic circle are considered as evidence of polar action, there can

* Loc. cit.

Loc. cit. Tom. VII. p. 20.

+ Experimental Researches, Vol. 11. p. 51.

§ Phil. Trans. 1852, p. 279.

be no reason why it should not be so considered in regard to organic action, viz. during secretion; but before we arrive at this conclusion let us compare the phenomena of secretion with another class of facts, viz. with those of osmose.

Professor GRAHAM has communicated a very valuable paper to the Royal Society, entitled On Osmotic Force, which has lately appeared in their Transactions*. In this paper Professor GRAHAM has shewn that osmose is dependent upon chemical action, and not as it has been generally supposed, upon capillary attraction. Time will not allow me to enter upon the facts brought forward in support of this opinion, and I must therefore refer to the paper itself, which cannot be too strongly recommended.

The conditions under which an osmotic experiment is conducted, viz. the necessity of having two fluids, one on each side of the septum, render it extremely difficult to ascertain by means of the galvanometer the exact mode of action which arises during osmose, so as to compare it with that which takes place in the animal body during secretion, in consequence of the reaction of the two fluids upon each other producing their own peculiar effects on the galvanometer; and the changes upon which osmose depends take place, according to Professor GRAHAM, within the substance of the porous diaphragm, where we cannot apply the electrodes of the galvanometer.

The fact of osmose depending upon chemical action shews however that the act itself must not be considered as a mere transudation, a mere physical separation, but that it depends upon other important conditions; and if upon chemical action they are consequently polar in their nature. If this conclusion be arrived at in regard to osmotic phenomena we may with equal propriety consider the phenomena connected with secretion to be at least something more than a mere physical transudation; and as reasons exist for shewing that osmotic phenomena are polar in their nature, why may we not consider the action connected with secretion, and where we can obtain such direct evidence of polar action, as manifested by the galvanometer, to be polar in their nature also?

Respecting the chemical character of osmose, and its bearings upon physiology, Professor GRAHAM adds:-"It may appear to some that the chemical character which has been assigned to osmose takes away from the physiological interest of the subject in so far as the decomposition of the membrane may appear to be incompatible with vital conditions, and osmotic movement confined therefore to dead matter. But such apprehensions are, it is believed, groundless, or at all events premature. All parts of living structures are allowed to be in a state of incessant change-of decomposition and renewal. The decomposition occurring in a living membrane, while effecting osmotic propulsion may possibly be of a reparable kind. In other respects chemical osmose appears to be an agency particularly well adapted to take part in the animal economy."

The subject of the present communication has been that of ORGANIC POLARITY, and to this it has been my endeavour to confine our attention, and to shew that some of the organic actions which occur in the animal body, viz. secretions, are evidently accompanied with the manifestation of current force; a fact which may not be disputed. An endeavour has been made

Phil. Trans. 1854.

also to point out with what class of phenomena they appear to be the most nearly allied, viz. those which occur in voltaic decomposition (a conjecture already advanced by WOLLASTON); and as these are considered as polar in their nature we are justified in logically inferring that those which occur in the animal body are likewise polar in their nature; and as chemical force is considered a polar force, so may organic force be viewed in the same light as a polar force also. But the conditions under which polar phenomena are manifested, in the organic, at once stamp them as of a higher order than those which are observed in the inorganic kingdom of nature.

Cambridge, Feb. 1858.

XIV.

A proof of the Existence of a Root in every Algebraic Equation: with an examination and extension of Cauchy's Theorem on Imaginary Roots, and Remarks on the Proofs of the existence of Roots given by Argand and by Mourey. By AUGUSTUS DE MORGAN, F.R.A.S., of Trinity College, Professor of Mathematics in University College, London.

[Read Dec. 7, 1857.]

To those teachers who value the logic of mathematics it has always been a subject of regret that the fundamental proposition of the theory of equations-every algebraical equation has as many roots as dimensions, and no more—is either to be taken on trust, or deferred to a late period of the course. Every such proceeding is, in mathematics, a confession of incompetency, either in the state of the subject or in the teacher. This confession I have until now been obliged to make by deferring the proof of the theorem until it can be deduced from Cauchy's theorem on the limits of imaginary roots, a theorem which incidentally brings out the existence of the roots. Having been recently led to examine the first* of Sturm's demonstrations of this theorem, in the first volume of Liouville's Journal, it struck me, from the very fundamental character of this proof, that there must be some equally fundamental demonstration of the existence of the roots, which would be the natural prefix to Sturm's demonstration. Attentive examination proved my conjecture to be correct; and at the same time I found an addition to Cauchy's theorem, which makes it include roots derived from the circuit itself, and also roots of the reciprocal of the function in hand. This I shall incorporate with Sturm's proof in the present paper: joining with it the consideration of Argand's and Mourey's proofs, which have points worthy of particular attention.

+, -,

The proof which I prefix to Sturm's demonstration depends upon a preliminary theorem, which is one of combination and position. It takes no account of the meaning of 0, ∞, + but only postulates that and shall be separated either by 0 or by All changes consistent with this condition are to be held allowable. Then + 0 + may become ++: but Either 0 or +0 — must not become + ∞ may open; that is, 0 +0, or 0 [+ 0 + 0-10 &c. Again + may become +0+or++; and so on. and may come together, and either cross each other or recede from each other without crossing; having, after crossing or recession, either the same sign between them as before, or a different sign.

0

may

become o

0, or And o

THEOREM. In any number of signs, each of which is + or -, interspersed with the signs o and ∞ in any manner which satisfies the condition that either 0 or ∞ always comes beand +, let k be the number of occurrences of +0 tween and and the number of occurrences of 0 +. Then it is impossible that I should undergo

and between

k

⚫ I mean the first by Sturm alone: the first in the memoir cited is by Sturm and Liouville jointly.

any alteration, unless by 0 and coming together, whether with change of place or simple recession. It is supposed that the series both begins and ends with a sign + or -, which remains unaltered; not with 0 or ∞.

Except appulse of 0 and, the only other changes are appearance or disappearance of 0 between like signs, appearance or disappearance of between like signs, opening of 0 or ∞ into 00 or ∞ ∞ with signs between them. A simple induction will shew that, in every case which involves no appulse of 0 and∞, either k and remain unaltered, or receive the same increment.

Thus when +0+ changes into ++, both are unaltered: as also in - 0 + changed into 0-0+, or - 0 - changed into 0 [0-]0. But in +0+ changed into + [0 - 0] +: +, both and increase by a unit: in +0 changed into +0 [-0+0]0-, both receive a +0[unit of increase. But when is changed into 0[+]0-, in which case k aug- 0 ments by a unit, while I is unchanged, the change, if continuous, commenced by an appulse of O and ∞, as in 00. Again, when 0++ changes through - - 0 + to - ∞ +0+, in which case / loses a unit, there is an appulse of 0 and ∞. This theorem brings the fundamental theorem on the roots of equations to rest on what will readily be acknowledged to be its proper foundation, the necessity of 0 or in the transition from positive to negative.

− 0 +, + ∞ —,

0+ ∞

Now suppose a line of any sort drawn in a plane, and at each point of it, (x, y), let the sign of a given function of x and y be recorded; with the character of each change, + 0 −, ∞, as the case may be. Every contour, and every portion of a contour, will thus present what we may call a chain of signs, such as + 0 with re0+ ..." ference to any function of x and y which may be chosen. If the contour, or part of a contour, change continuously, so as to pass gradually from one form and position to another, changes may occur in the chain; and it is obvious that the change may be so conducted, that not more than one of the signs 0 and ∞ shall be affected at any one moment. If the P where P and Q never become infinite for any finite values of x and y, then O can only appear when P = 0, and of 0 and

function examined be

Q'

P

can only take place where

Q

can only appear when Q = 0, and an appulse

0

0

takes the form. Next, suppose p≈ to be a function which never becomes infinite for any finite value of z, and let p(x + y − 1) = P+Q-1. We see then that if k-1 be found to have, on one contour, a value different from what it has on any other contour, a gradual transition from one contour to the other cannot be made without the varying contour passing through points at which P= 0, Q = 0, or f (a) = 0. Such point or points then must exist; or we have the following

THEOREM. If ƒ (x + y √ − 1) = P+Q-1, and if neither P nor Q can be infinite for any finite values of x and y; if also two contours can be found for which k - l has different values; then such difference of value is proof of the existence of a root or roots which satisfy z = 0.

It is supposed that the choice begins and ends with fixed signs. This always takes place when we go round the whole of a closed circuit, from one sign to the same again. But we have also seen that, so long as the initial and terminal signs remain the same, it is impossible