The Deputation appointed by the Council to represent the Society on the occasion of the 400th anniversary of the founding of the University of Upsala consisted of Mr Sprague and myself. Professor Balfour and Professor Sir Wyville Thomson, who represented the Edinburgh University, also joined the Deputation.

The transformations of volume integrals into surface integrals, and of surface integrals into line integrals, are usually established first for a small parallelepiped and parallelogram. These elements are not, however, very convenient, or indeed altogether satisfactory, for the extension of the theorems over finite domains. If arranged in straight rows and plates, the elements cannot be fitted to a general boundary; and if they are arranged in curved rows and shells like the stones of a vault, there will be, in general, between various elements of the first order, spaces of the second not represented. This inconvenience disappears if tetrahedra and triangles are taken as elements. In Quaternions, the theorems are easily proved for such elements.

We present new estimates in the setting of weighted Lorentz spaces of operators satisfying a limited Rubio de Francia condition; namely $T$ is bounded on $L^{p}(v)$ for every $v$ in an strictly smaller class of weights than the Muckenhoupt class $A_p$. Important examples will be the Bochner–Riesz operators $BR_\lambda$ with $0<\lambda <{(n-1)}/2$, sparse operators, Hörmander multipliers with a limited regularity condition and rough operators with $\Omega \in L^{r}(\Sigma )$, $1 < r < \infty$.

(1) It is generally assumed that the vacuum photo-cell has no appreciable time lag. The electrons start out from the surface as soon as the light falls on it. The most important investigation on the subject is that of Marx and Lichtenecker (1913). By means of a mirror rotated 170 times per second an image of a narrow slit was swept over the cathode of a photo-cell at a distance of 8·3 metres from the mirror; the cathode was a potassium one sensitised with hydrogen. The electrometer reading was noted, and then the mirror was rotated at a much slower speed and the electrometer read again. No appreciable difference was recorded. In the words of the authors an alteration of the time of illumination from 4·38 × 10-3 to 1·46 × 10-7 sec., the intensity of radiation being 0·56 erg/sec. cm.2, had no effect on the result to an accuracy of one or two per cent. Theoretical deductions have been drawn from this statement which have been much quoted. It seemed to me that the statement itself was worth verifying, and this paper is the record of an attempt to verify it and push the limit of accuracy further.

Referring to the nerves of touch, as distributed to the points of the fingers, the author proceeded to shew that something more was necessary than the exposure of the nerve. For example, the tongue being the most perfect organ of touch, resulting from the fine supply of nerves, is yet incapable of receiving certain impressions;—so that although it could distinguish the finest hair, so fine as not to be felt by the finger, yet if applied to the artery at the wrist, it conveys no sense of pulsation.

Colin Maclaurin, the celebrated mathematician, was born in 1698 at Kilmodan in Argyllshire, where his father was minister of the parish. In 1709 he entered Glasgow University, where his mathematical talent rapidly developed under the fostering care of Professor Robert Simson. In 1717 he successfully competed for the Chair of Mathematics in the Marischal College of Aberdeen University. In 1719 he came directly under the personal influence of Newton, when on a visit to London, bearing with him the manuscript of the Geometria Organica, published in quarto in 1720. The publication of this work immediately brought him into prominence in the scientific world. In 1725 he was, on the recommendation of Newton, elected to the Chair of Mathematics in Edinburgh University, which he occupied until his death in 1746.

Convergence theorems for the practical eigenvector free methods of Gay and Goerisch are obtained under a variety of hypotheses, so that our theorems apply to both traditional boundary-value problems and atomic problems. In addition, we prove convergence of the T*T method of Bazley and Fox without an alignment of projections hypothesis required in previous literature.

For Q ∊ L2(0, 1) we investigate the set Γ ∊ ℝ2 of pairs (α β,) for which the problem

has a nontrivial solution which has exactly one zero in (0,1) and is positive near x = 0. We show that Γ is stable in a certain sense under small perturbations of Q. The dependence of Γ upon Q is illustrated by an example.

We give local results related to Hopf bifurcation for parabolic equations. The linear part about the equilibrium point can have zero eigenvalues. In our results the information about the perturbation is essential and it is possible to obtain bifurcation even if some ‘i’ or zero remains on the imaginary axis for all values of the parameter.

We consider the class of nonlinear eigenvalue problems where yp* = |y|p sgn y, pi > 0 and p0p1 … pn−1 = r, with various boundary conditions. We prove the existence of eigenvalues and study the zero properties and structure of the corresponding eigenfunctions.

In this paper, we are concerned with discrete Schauder estimates for solutions of fully nonlinear elliptic difference equations. Our estimates are discrete versions of second derivative Hölder estimates of Evans, Krylov and Safonov for fully nonlinear elliptic partial differential equations. They extend previous results of Holtby for the special case of functions of pure second-order differences on cubic meshes. As with Holtby's work, the fundamental ingredients are the pointwise estimates of Kuo-Trudinger for linear difference schemes on general meshes.

We study a Ginzburg–Landau functional for a vector-valued order parameter that carries a spin and couples directly to the magnetic field. We verify that spin coupling reduces the lower critical field Hc1, the smallest value of the applied field strength at which vortices penetrate the superconductor, and discuss the related phenomena of a ‘spontaneous vortex phase’ for sufficiently strong spin coupling, predicted by physicists. The method uses energy methods and a priori bounds to confine the vortices and spin transitions to very small balls

The ring of a finite projective plane, introduced by the author in a previous paper, is shown to be a Gorenstein ring. Its integral closure and conductor are also determined.

A saddle-point property of the self-similar solutions in the curve shortening flow was conjectured by Abresch and Langer and confirmed by Au. An improvement on Au's solution is presented.

A characterisation is obtained of all the regularly solvable operators and their adjoints generated by general ordinary quasidifferential expressions in The domains of these operators are described in terms of boundary conditions involving the solutions of M[u] = λwu and the adjoint equation at both singular end-points a and b. These results are an extension of those proved in [3], by Evans and Ibrahim, to the case of two singular end-points of the interval (a, b), and a generalisation of those in [10] and [13] concerning selfadjoint and J-selfadjoint differential operators, where J denotes complex conjugation.

During the last fifty years not a few familiar and apparently quite uninteresting structures have all at once assumed unusual importance and arrested the attention of naturalists in all parts of the world. This is especially true of what used to be known as rudiments, of what we nowadays generally designate vestiges. Of all the known vestiges, the most familiar are perhaps the “splint” bones of the horse. For long the interest in these splints was extremely limited, but since it was shown that they correspond to the functional second and fourth metacarpals and metatarsals of other vertebrates they have attracted the notice of both comparative anatomists and palæozoologists, and are now invariably looked upon as affording strong evidence in support of the view that the horse has descended from polydaetylous ancestors.

In a previous paper the hypothesis propounded by Kapteyn that the stars are moving in two streams was examined by considering the proper motions of stars which were moving more than 20″ a century. By a graphical method the directions of these two streams were found with results in fair agreement with those found by Kapteyn and Eddington by other methods and with different material. It was evident from that paper that it would be possible in a large percentage of cases to say with tolerable certainty to which stream individual stars of large proper motion belonged.

The questions to be resolved by a complete theory of the motive power of heat, are the following:—

I. What is the precise nature of the thermal agency by means of which mechanical effect is to be produced, without effects of any other kind?

II. How may the amount of this thermal agency necessary for performing a given quantity of work be estimated?

The equation studied here is Lny +p(x)y = 0, where Ln is a disconjugate differential operator and p(x) is of a fixed sign. It is shown that certain minors of the wronskian of this equation satisfy a very similar differential equation Mnz +p(x)z =0. We prove that some properties of the original equation which are essential for oscillation theory are inherited by the solutions of the second one.