We consider damped nonlinear hyperbolic equations utt + Aut + αAu + βA2u + G(u) = 0, where A is a positive operator and G is the Gateaux derivative of a convex functional. We examine the asymptotic behaviour of solutions and the convergence of strong solutions to these equations when the parameter β tends to zero.

We investigate critical points of the functional

over a ball in ℝn. Here, W is radially symmetric but not convex. We embed the functional into a family of functionals

where E0,0(u) = E(u). A global bifurcation analysis yields a branch of non-trivial critical points depending on λ and positive ε, where we can set λ = 0. The geometric properties preserved on that branch, due to the maximum principle, prove compactness such that the singular limit as ε ↘ 0 exists. Under natural conditions on W and G the critical point obtained in this way is a minimizer of the original functional. That plan can be carried out only under the restriction of radial symmetry, since the maximum principle applies only to special elliptic equations of fourth order. That restriction, however, is not essential since every minimizer of the functional is radially symmetric.

Suppose that f: ℝ×ℂN→ℂN is holomorphic in z and continuous in t, and that Φ: ℂN×ℂN→ℂN is holomorphic. Boundary value problems of the form

are considered. The particular interest is in the structure and topological properties of the set of solutions. The paper is motivated by the corresponding properties of the set of periodic solutions of ż = f(t, z) when f is periodic in t. Consideration of this complex equation gives information about the periodic solutions of the real equation ẋ = f(t, x).

Professor R. D. Brown has brought to our attention an error in our Theorem 4.2. This would, in turn, invalidate Theorem 3.6 which depends on Theorem 4.2. The basic flaw is that the alignment hypothesis of Proposition 4.1 is not satisfied by the product space projections, Qn. Interestingly, such product space constructions are a common computational device in applications of the T*T method, but apparently seldom satisfy the alignment hypothesis of Proposition 4.1. We wish here to correct these theorems, and comment on the resulting implications for Example 6.1. In independent and as yet unpublished work, Professor Brown has obtained further new results on convergence of the method of intermediate problems.

Braier (1931) showed that hypophysectomised dogs on a complete diet excrete 24 per cent. less urinary creatinine than the controls, and 35 to 40 per cent. less when fasting. Schrire and Zwarenstein (1933, 1934) found that injection of anterior lobe extracts increased the urinary creatinine in rabbits.

These results afford evidence of an endocrine relationship between the anterior lobe of the pituitary and urinary creatinine. In view of the well-established relationship between urinary creatinine and muscle creatine the present authors decided to investigate the relationship of the anterior lobe to muscle creatine. An effect on phosphocreatine of muscle was reported by Marenzi (1933), who obtained a decrease of 33 per cent, following removal of the anterior lobe of the pituitary. The effect of hypophysectomy and of injections of anterior lobe extracts on muscle creatine has not been previously investigated.

Most of the work on the emission spectra of the halogens has been confined to the visible or near ultra-violet regions: but, in view of the characteristic chemical and photo-chemical properties of these elements, it has become increasingly important to make a general survey extending well into the ultra-violet. The emission spectra of chlorine and bromine in the visible and near ultra-violet up to about 370 µµ have been described by Eder and Valenta as consisting of lines, chiefly in the green and blue. For bromine, a continuous spectrum was also observed, starting at about 490 µµ; this rapidly falls off in intensity, but at about 420 µµ a second continuous band commences, extending with increasing intensity to the limit of their photograph about 370 µµ. Quite recently the investigation of the quartz ultra-violet region of chlorine has occupied several observers, chiefly from the physical point of view; Jevons has measured the ultra-violet line spectrum from 335 µµ to 207 µµ, and a communication on the same subject has appeared from Angerer.

The theorem itself may be considered obvious, and is easily applied, as I showed at the late meeting of the British Association, to prove that in passing from any one double point of a plane closed curve continuously along the curve to the same point again, an even number of intersections must be passed through. Hence, if we suppose the curve to be constructed of cord or wire, and restrict the crossings to double points, we may arrange them throughout so that, in following the wire continuously, it goes alternately over and under each branch it meets. When this is done it is obviously as completely knotted as its scheme (defined below) will admit of, and except in a special class of cases cannot have the number of crossings reduced by any possible deformation.

The Nudibranchs owe their prominence in British zoology to the late Dr Johnston of Berwick, and to the splendid monograph of Messrs Alder and Hancock.

Inhabitants for the most part of the laminarian and littoral zones, the rocks and rock-pools, by minute and continued search, produced the greater number of those met with at St Andrews, the few others being procured from fishing-boats and debris of storms. Almost all require to be sought for with care, and generally escape superficial notice altogether. Most have been kept for a longer or shorter period in confinement, and some are living even now, though with greatly diminished bodies, and less brilliant tints, the results of insufficient food and other unfavourable circumstances.

When an elastic body is constrained in any manner whatsoever, it is susceptible to vibration, by virtue of its elasticity, when disturbed from its position of equilibrium by an externally applied force. The period and amplitude of such vibration are dependent upon the mass and inertia of the system, the rigidity of the constraints, and upon the nature of thedisturbing force.

When a beam of commercial section is loaded centrally, and subjected to vibrations, the frequency of transverse vibration can be readily determined from a knowledge of the dimensions of the beam, its modulus of elasticity, and the conditions of loading. On the other hand, where the loading is eccentric, the transverse vibration is accompanied by a torsional vibration the frequency of which is very much lower than is indicated by the ordinary elastic theory, due to the inefficiency in torsion of beam sections other than circular. In practice it is not always possible to eliminate the eccentric loading of beams, as for instancewhere power is transmitted through countershafts supported from structural steel-work, and it is with the problem of the torsional vibration of such eccentrically loaded beams that itis proposed here to deal.

The present memoir is a continuation of those already published by one of us, on the physiology of the Invertebrata.

In 1907 I had the honour of reading a paper before this Society in which a new genus of iron-bacteria was described. This organism was first discovered in Renfrew, near Glasgow, and subsequently observed in various samples of iron-water from different parts of the country. At Renfrew this organism was at the time of its discovery the sole bacterial occupant of the ironwaters of this neighbourhood. In most samples of these waters—and I have examined hundreds—only the remains of iron-bacteria can be obtained, but occasionally the samples are found to contain iron-bacteria in process of growth and multiplication. After a little experience, it is possible to ascertain which streams contain iron-bacteria in this favourable condition. At Renfrew such streams were then common, and so gave me an opportunity of tracing the life-history of this species. The results of this investigation seemed to me to justify the conclusion that I had obtained a new genus of iron-bacteria.

A differential equation may be regarded from two points of view, one purely analytical, the other geometrical. From the analytical point of view, a differential equation of the first order is merely a functional relation between x, y, and p (where p = dy/dx), and the problem of solving the equation is to find a function of x, say f(x), such that if f (x) and df(x)/dx are substituted for y and p in the equation, the result is an identity in x. In the geometrical method, on the other hand, x and y are treated as the co-ordinates of a point in a plane and p as a direction. The differential equation then attaches to every point in the plane a certain direction, which may be conveniently represented by an arrow drawn through the point. The problem of integration then resolves itself into finding a family of curves, such that, at every point (x′, y′), the direction of the curve at that point is the direction obtained by substituting x′ and y′ in the differential equation and solving for p. These curves are called the integral curves of the equation. This method owes its development chiefly to Lie.

In this paper are collected and discussed the observations of the specific gravity of saline solutions of moderate and high dilution made during the course of the last ten years. The instrument employed was the hydrometer, of the type used by me during the cruise of the Challenger in the years 1872 to 1876. The results then obtained were such as to convince me that the instrument, in capable hands, and with strict attention to the obligatory conditions of experiment, gives more exact results than any other instrument in use for such work. In the course of the last forty years the method has been improved in the sense that more and more attention has been paid to the conditions, especially those of temperature, under which the experiment is made, because it was found that the method is so delicate that it responds to every increase of care in experimenting by greater exactness in the results which it furnishes.

The alteration of the amount of hydrolysis by change of temperature has been directly observed in connection with various salt solutions. In the case of ferric salts, quantitative determinations have been carried out by Wiedemann, employing magnetic methods. He showed that, in the case of the sulphate, the amount of hydrolysis is considerably increased by rise of temperature. A similar effect is observed in the case of gallic sulphate, and here the change is directly evident; a solution of the salt when heated deposits white basic salt, which re-dissolves on cooling. A somewhat analogous case which has recently come under my notice, and which, so far as I am aware, has not been noted previously, is provided by thallic sulphate solution. The thallic salts generally are decomposed more or less easily by water, with formation of a brown precipitate, but dissolve in dilute acids, giving colourless solutions (Crooke, Lamy, Willm, Strecker). An acid solution of thallic sulphate, provided it does not contain too great an excess of sulphuric acid, may be used to illustrate several points connected with the hydrolysis of salts. Dilution with water produces a brown precipitate. A similar precipitate is formed in considerable quantity when the solution is warmed; when the mixture cools, the precipitate slowly re-dissolves, and a clear colourless solution is again obtained. It is therefore evident that the formation of the precipitate is not due simply to a colloidal substance being rendered insoluble, but to a greatly increased hydrolytic action.

In the third Book of The Wars of the Jews, Chap. VIII. § 7, we are told that Josephus managed to save himself and a companion out of a total of 41 men, the majority of whom had resolved on self-extermination (to avoid falling into the hands of Vespasian) provided their leader died with them. The passage is very obscure, and in a sense self-contradictory, but it obviously suggests deliberate fraud of some kind on Josephus' part.

“And now,” said he, “since it is resolved among you that you will die, come on, let us commit our mutual deaths to determination by lot. He whom the lot falls to first, let him be killed by him that hath the second lot, and thus fortune shall make its progress through us all; nor shall any of us perish by his own right hand, for it would be unfair if, when the rest are gone, some-body should repent and save himself.” Whiston, Works of Flavius Josephus, IV. 39.

Bachet, in No. XXIII. of his Problèmes plaisants et délectables, makes a definite hypothesis as to the possible nature of the lot here spoken of; so that the problem, as we have it, is really his.

“Supposons qu'il ordonna que comptant de 3 en 3 on tuerait toujours le troisième, …. il faut que Josèphe se mît le trenteunième après celui par lequel on commençait a compter, au cas qu'il visât à demeurer en vie lui tout seul.

It is shown that the free product of two residually finite combinatorial strict inverse semigroups in general is not residually finite. In contrast, the free product of a residually finite combinatorial strict inverse semigroup and a semilattice is residually finite.

The length of the old English mile has been hitherto so uncertain, that any fresh light upon it is well worthy of study; and an important source of information—the map in the Bodleian Library— has not yet been brought to bear upon the question. The present inquiry was suggested by the sight of this map, which seems to add so much to our knowledge that a review of the whole subject has become desirable. It is proposed, therefore, in this paper to bring together all the data worth consideration, beginning with the most recent; and by deducing what the mean conclusion is from each source, to be thus able to compare together the various results, and so arrive at some definite statements within known limits of uncertainty.

The recognition of the importance of lateral strain in the theory of elasticity, as now taught in most engineering colleges, makes it very desirable that students should make experiments upon the lateral contraction of tension specimens and the lateral extension of compression pieces with the same facility that they now determine the values of Young's modulus and the modulus of shear.

On the 16th of February 1880 I communicated to the Society a research upon Comets, which led me to believe in the existence of two new planets outside the orbit of Neptune. This memoir was not published in the “Transactions,” and only a short abstract appeared in the Proceedings of the Society. A small number of copies were privately printed; but any one wishing for information on the subject will find it most readily in the “Observatory,” 1880, June 1.

The decided tendency of the aphelia of comets to group themselves at distances from the sun equal to those of the planets could leave no doubt about the existence of these two new planets. The only doubtful point was as to the positions of the planets in their orbits. Meantime this research gives us greater confidence in the accuracy of the orbits of comets which have been calculated. This is most especially the case with the periods of revolution of comets when those periods are long, an element which has hitherto been very generally mistrusted. The most remarkable case is that of the comet 1861, which was calculated by Oppolzer to have a period of 415 years, and which I showed to have returned. three times at such an interval.