In this paper we give sufficient conditions (Theorems 1 and 2) for the instability of the fourth order differential equation

An Lp inequality for l < p < 2 is established and applications to fixed points of uniformly Lipschitz mappings and strongly unique best approximations are given.

Suppose that f and g are transcendental entire functions such that the composition F = f(g) has finite order, and suppose that Q is a nonconstant rational function. We show that N(r, 1/(F – Q)) ≠ o(T(r, F)). The theorem is related to results of Bergweiler, Goldstein and others.

Let denote a subring of the complex field that contains 1 and is closed under complex conjugation. It is shown that, with respect to the involution induced by word-reversal, the algebra over of a free monoid admits a trace and a separating family of star matrix representations. From the existence of a trace it is deduced that the aforementioned involution is special, in the sense of Easdown and Munn. Similar results hold for the algebra over of a free monoid with involution.

For β > β0: = 1 −[(n − 2)/2]2 and n ≧ 2, it was recently shown by Simon that the self-adjoint operator associated with −Δ + βr−2 in L2(ℝn) has domain H2(ℝn) ∩D(r−2) the constant β0 being the best possible. An alternative proof of this result is given.

Let d(L) denote the deficiency indices (which are equal) of a formally symmetric differential expression L with real coefficients. Then it is shown that

(a)if k>1, d(Lk)−d(Lk−1) ≤ order L, and

(b)if j ≥ k > 1, d(Lj)−d(Lj−1) ≥ d(Lk)−d(Lk−1).

In this paper we consider the problem of defining a convolution, analogous to the classical convolution of scalar measures, of measures defined on the Borel sets of a locally compact semi-group S and having values in a Banach algebra . Using the bilinearintegral introduced by Bartle we show that the formalism of the scalar case persists in situations of considerable generality so that the formula

suitably interpreted, gives a Banach algebra structure to a large class of valued measures defined on S. Themethods exploit the connection between vector measures and operators and involve some results of independent interest.

Some time ago, while studying the chemical dynamics of the changes which occur in solutions of urea or carbamide, I came upon some rather unexpected results which led me to hope that investigations conducted on somewhat the same lines with other substances of the amide group might prove to yield results of some interest. The amides referred to are those derived from carboxylic acids. While proceeding to this investigation I noticed some measurements, obtained in connection with the viscosity of aqueous solutions of carbamide, which appeared of sufficient interest to demand an inquiry into the nature of solutions of this class of substances before proceeding further with the subject of inquiry in the manner at first intended.

Zeleny (Physical Review, xii., p. 321, 1901) has investigated the variation with temperature of the photo-electric discharge from a platinum wire in air at ordinary atmospheric pressure.

The object proposed by the author of this paper, was to present, in a complete form, the results of investigations into the phenomena of Waves, in which he had been engaged for several years. A part of the experiments referred to in the paper, had been carried on by Sir John Robison, conjointly with the author of this paper, as a Committee of the British Association. The others were new, and had been carried on by himself, for the purpose of completing and giving that systematic form to our knowledge of the subject, which was attempted in this paper.

During the deep sea dredging expedition of H.M. ships ‘Lightning’ and ‘Porcupine,’ in the summers of 1868–69 and 1870, two or three nearly perfect specimens, and a number of fragments were procured of three species of regular echinideans, which were referred by the author to a new family, the Echinothuridæ, intermediate in their more essential characters between the Cidaridæ and the Diadematidæ.

In these urchins the test is circular and greatly depressed. The plates of the perisom are long and strap-shaped, and the interambulacral plates overlap one another regularly from the apical towards the oral poll, while the ambulacral plates overlap in a similar way in the opposite direction.

In Kayser's Spectroscopie, vol. iii. pp. 45–49, an account is given of all the different methods that have hitherto been employed for photometry in the ultra-violet part of the spectrum. Photography, phosphorescent plates, selenium cells, and the ionising effect of ultra-violet light have all been used, but with limited success, and Professor Kayser considers that the only really practical method is that which has been recently introduced by Pfluger. Pfliiger has discovered that there is relatively an enormous amount of energy in the ultra-violet spectrum of the electric spark produced by the discharge of a condenser between metal electrodes. He therefore uses the electric spark as a source, and takes the deflections with a thermopile and galvanometer, just as in the infra-red. He finds the deflections to be wonderfully steady, considering the inconstant nature of the spark.

J'ai reçu par une voie détournée, un article de M. Edward Sang, intitulé “Remarks on the great Logarithmic and Trigonometrical Tables computed in the Bureau du Cadastre under the direction of M. Prony.” Cet article, qui parait avoir été publié dans les “Proceedings of the Royal Society of Edinburgh, Session, 1874–1875,” m'a été adressé à l'Observatoire de Paris. Or je n'ai pas l'honneur d'être astronome. Je suis simplement un inspecteur général des Ponts et Chaussées, ami de la science qu'il a cultivée et qu'il cultive encore dans les trop courts instants de loisir que lui laisse sa carrière professionelle. L'Observatoire de Paris a d'ailleurs bien voulu me reiivoyer la brochure à mon domicile, rue du-bac No. 38, à Paris.

In the former parts of this investigation we studied the physiological action of the salts of a considerable number of ammonium bases—that is, of the salts formed by the union of an ether with the nitride of one or more alcohol radicals. Thus—

The examination of the physiological action of such salts proved that, while differing from one another in many respects, there are two points in which they agree—they all paralyse the end-organs of the motor nerves, and none of them possess that stimulating action of the spinal cord which we observe in such a substance as strychnia.

The apparatus used in the following determinations is that of Victor Meyer, modified as described in a paper published in 1879, “On the Vapour Densities of Potassium and Sodium,”* by Professor Dewar and the author. In the experiments here described, the platinum vessel was further protected by a casing of iron, and the intervening space filled with magnesia, the iron casing being embedded in sand enclosed between two crucibles.

The position taken by Mr Darwin is, that all species have arisen by the natural process of ordinary generation. That the differences which we now see in them have arisen from slight variations in individuals having from time to time occurred, which have been perpetuated by inheritance, by successive stages and slow degrees, through unlimited spaces of time. Some of these slight variations he considers to originate in causes beyond our power of explanation, and which, although not the work of chance, we may call chance, for want of a better appellation—others to arise from habit, or from the excessive use or disuse of certain organs; but that when such a variation has once appeared, it is preserved by hereditary descent through a principle which he calls “natural selection,” and which he deduces as a corollary from the struggle for existence which we see constantly going on around us.

From M. Lefort's opening sentence it appears that he had only recently received the copy of my remarks which had been posted to him on 22d December. Perhaps on this account M. Lefort has been hurried in the perusal of my paper, and so has fallen into several mistakes as to my meaning. These will be apparent to any one who peruses the writings, and I shall pass them over entirely, confining myself to the very few points which are essential to the subject in hand. The only extraneous matter to which I shall allude is this, that while M. Lefort has obviously and justly been desirous of upholding the dignity of the Grandes Tables du Cadastre, he has, in the true spirit of an inquirer after truth, clearly and faithfully exhibited even those points which press most sorely on his own position.

The fact that the temperature of the earth increases with the depth below the surface, implies a continual loss of heat from the interior by conduction outwards, through or into the upper crust. Since the upper crust does not become hotter from year to year, there must therefore be a secular loss of heat from the whole earth. It is possible that no cooling may result from this loss of heat, but only exhaustion of potential energy, which in this case could scarcely be other than chemical affinity between substances forming part of the earth's mass. But it is certain that either the earth is becoming, on the whole, cooler from age to age, or that the heat conducted out is generated in the interior by temporary dynamical action (such as chemical combination).

The subject of this paper was studied by Orr (Camb. Phil. Trans., vol. xvii, 1898, pp. 171–199; 1899, pp. 283–290), and later by Barnes (Proc. Lond. Math. Soc., ser. 2, vol. v, 1906, pp. 59–116), in whose paper a number of references to earlier work on the subject are given. Formulæ equivalent to (9), (18), and (20) below were given by, Barnes, who derived them by his well‐known method of integrating products and quotients of Gamma Functions. In this paper the formulæ are deduced by induction from simpler formulæ which are established in section 2. In order to simplify the notation, four functions, denoted by P, E, Q, and H, are introduced, the first two in section 3, the last two in section 5. The P and Q functions are merely generalised hypergeometric functions multiplied by convenient factors. The E function, which is equivalent to Barnes's contour integral, is defined as a multiple integral, and from it the asymptotic expansion, with a useful form for the remainder, is easily derived. The H function is a multiple of the E function.

This paper, following Stratford [9], considers a boundary layer developing with uniform pressure when x < x0, and with large and constant when x > x0. The analysis improves or extends the work of Stratford as follows.

(a) For large values of inner and outer asymptotic expansions are derived and matched—Stratford considered only the inner solution, his outer boundary condition being determined by physical arguments.

(b) The skin-friction is determined as a series in powers of , whereas Stratford's, solution was obtained by approximate methods. In the limit as λ→∞ the predicted separation position, ξ = 0.09766, is probably correct to3 significant figures, at least, and differs little from Stratford's value.

(c) This solution further shows that for finite λ, separation occurs when

ξ = 0.09766-0.00014λ−3.

Although the coefficient 0.00014 is not wholly reliable, clearly the position of separation varies little with λ unless λ−3 is very much greater than unity.

(d) The present solution also yields the displacement and momentum thicknesses. When x > x0 the displacement thickness is

where B1(ξ) is given as a slowly converging seriesin powers of ξ. It is shown how to improve convergence, using certain properties of the flow close to the separation position, and thus to sum the series.

Likewise

so that when λ is large the momentum thickness changes even less than does the displacement thickness.