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A graph paper on which a Gaussian function of unit amplitude appears as a straight line was produced in an attempt to devise a method which would combine the obvious advantages of a linear presentation with the improved definition and convenience of a differential plot. A semilogarithmic plot of the linear differential pulseheight spectrum of 622 kev gamma ray from a Cs137 source is given as an example of the technique. The method requires only the data points from the multichannel analyzer and an estimate of the maximum height of the photopeak above background. Gaussian plots proved convenient checks on instrumental reliability; amplifier nonlinearity or drift cause noticeable deviations from linearity. (P.C.H.)

A part of the dye-protein complex formed in paper electropherograms of blood sera cannot be eluted from sections of the electropherograms by heating the sections with a N hydrochloric acid at 100°C. However, the noneluted part of the dye-protein complex may be calculated by assuming that it is adsorbed by the paper and therefore by applying the Langmuir isotherm equation.

The argument of this paper map be regarded as centred round the ascertainment of two linear differential equations, those denoted by (P′ - p ) η -1 Ψ = 0, (Q′ — q ) η -1 Ψ = 0; these are, in fact, differential equations satisfied by a quotient of theta functions of several arguments when all of these arguments except one are taken constant, and the theory of the paper is in close connection with the classical method for approaching the definition of the theta function from the algebraical side. It would seem to be important to justify this assertion in precise terms, so as to make clear the general bearings of the results arrived at by the authors. The elementary algebraic method which they follow has in my opinion great interest, especially as it enables them to deal in part with the theory even when the curve has unknown multiple points; but one is not satisfied until the functions involved are defined by their behaviour and not by their algebraical form. I hope, therefore, that the writers will allow me to make the following remarks; and that I may be pardoned for explicit references to my own volume, ‘Abel’s Theorem,’ 1897. This was witten before the appearance of Weierstrass’ lectures (‘Werke,’ vol. 4, 1902; 600 quarto pages), and is very imperfect; but it is briefer in one respect than Weierstrass’ theory, by the use of the fundamental integral functions (Kronecker, Dedekind and Weber, Hensel). What there is of novelty in the following remarks relates mainly to these. The general ideas are expounded also in Clebsch and Gordan’s ‘Abelsche Functionen’ (1866)—to which, as to Weierstrass’ volumes, explicit references are given below. I consider in the first place a curve of which the multiple points are known and allowed for; but exemplify in examples how the theory can be applied when this is not so, the existence of a new multiple point being regarded as a particular case.

Three cases and sources of inaccuracy in finding the value of E in the former paper (Proc. Roy. Soc. vol. xv. p. 429) having been pointed out by Mr. Glaisher, and some other minor errors not noticed by him having since been detected by the author, and these having vitiated the results, but only in a slight degree the inferences drawn from them (for in the former paper the last and leading conclusion as to the value of E, though limited , was certainly correct ), the author has been led, from a deep sense of obligation to the Royal Society, to revise, correct, and extend what he had previously done. And it will be seen, from comparing Mr. Glaisher’s remarks and results with what follows in this paper, that the supplementary matter herein given, including the extension of E &c. to 110 places of decimals, can scarcely be without interest to mathematicians, and especially as regards the summation of the harmonic series in the formula for finding the value of E. Not having seen M. Oettinger’s article in Crelle’s ‘Journal,’ “On Computing the value of E,” the author is unable to state what artifices he used in summing the harmonic series. Mr. Glaisher gives a very simple and obvious one from M. Oettinger, which the author could not but see and employ for calculating the values of the reciprocals of the even numbers.

Discussion of the problem of an elastic half-space with spherical cavity is continued in respect of Rayleigh waves on the plane boundary. Displacements in the initial and first group of higher order Rayleigh waves are derived by using the time-harmonic solution developed in part I of this series with attention confined to the case of time-harmonic normal stress at the cavity. These are employed to find also the response to an exponential shock at the cavity and graphs are presented showing the surface motion due to the initial Rayleigh waves. Finally, in an appendix to the paper, some comments are given on a recent paper by R. D. Gregory on the problem of the half-space with cavity.

This paper calls attention to an experimental design which enhances the efficiency of the technique described by B. F. Houston and R. A. Huffman [3]. This alternative design is a modification of the simplex lattice design developed by Scheffe [6] for the mixture problem. While not possessing the optimal statistical properties of the central composite design proposed in [3], such as minimum bias and minimum variance, this modified design is optimum with respect to the number of design points employed; hence, it is the minimum cost design. Further, the design proposed in this note is excellent for use in multiple-block experimentation, which is very often required with constrained systems.

Recently Cameron and Storvick introduced and studied an operator valued function space integral related to the Feynman integral. The main theorems of their study establish the existence of the function space integral as a weak operator limit of operators defined at the first stage by finite-dimensional integrals. This paper provides a substantial strengthening of their existence theorem giving the function space integrals as strong operator limits rather than as weak operator limits.

Let Em and En be orthogonal Euclidean spaces of dimensions m and n respectively and with the origin of each as their only common point. In a previous paper (3) I gave what was intended to be a proof of the relationwhere the dimension of A, dim A, is the Besicovitch dimension, i.e. the number s such that the Hausdorff measure in any dimension greater than s is zero whilst that in any dimension less than s is infinite, where A and B are subsets of En and Em respectively and where A × B is the Cartesian product of A with B.

A conducting paper analogue has been employed to provide a rapid and inexpensive method for the determination of values of stress in fields of uniaxial tension and of pure torsion, to which the equations of Laplace and Poisson apply. The reliability of the equipment has been verified by comparison of selected experimental results with those of known theories. Applications of the technique to the determination of the stress distributions in thin sheets containing geometric openings are given, and also to the determination of the stress distribution in a circular shaft containing a keyway, in pure torsion. The relevance of of Inglis' theory for equivalent elliptic holes is examined. Results are presented in graphical form.

The importance in Mathematical Physics of the Bessel functions, whose order is half an odd integer, suggests that the corresponding Mathieu functions may be worthy of a closer attention than they have yet received. At the very least it is expedient to pave the way for their computation. In the second paper bearing the above title, asymptotic developments of the characteristic numbers which correspond to these functions were given; it is here proposed, in the first place, to take the more direct line of approach.

We illustrate a technique for computing certain integrals that arise in probability theory by giving a new derivation of a formula of Kingman. This formula contains the joint distribution of the processes F(t) = inf {s: X(t + s) = b} and B(t) = inf{s: X(t - s) = b} where X is a time homogeneous, continuous parameter, Markov process and b is a fixed point in its state space. We then extend this formula to the situation in which b is replaced by a finite set {b 1, …, b n }.

The proof of one of the main results of C. N. Friedman, Perturbations of the Schroedinger equation by potentials with small support ( J. Functional Analysis 10 (1972), 346–360) is modified basing it on a more general existence theorem. A proof of this existence theorem is provided.

The aim of this short note is to improve some recent results of B. Grunbaum by some remarks. We use Grunbaum's notations.

The dynamic response of an infinite, elastic plate to a steadily moving transverse force is studied under the so-called improved plate theory. Analytic solutions for the subsonic and the intersonic cases are obtained using the method of integral transforms. The usual difficulty of the nonuniqueness of the steady-state solutions is circumvented by formulating the problems as initial value problems and then seeking the large time limits of the solutions of such initial value problems. The subsonic and intersonic solutions obtained here are similar to that obtained previously for the supersonic case. However, in contrast with the supersonic solution, which contains two families of Mach lines, the subsonic solution contains no Mach lines and the intersonic solution contains only one family of Mach lines.

A comparison of methods for the numerical solution of the hollow waveguide problem is presented in tabular form. Another table lists waveguide shapes and their cutoff characteristics that have been presented in the literature. These tables and the bibliography afford an aid towards the selection of a method.

By determining the germinating power of potatoes of the Bintje variety it could be clearly demonstrated that seed potatoes (also when stored without sprouts at 2 C)-undergo changes in their germinating power and exhibit senility symptoms about twelve months after being harvested.

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The theory of constructive formation of plane algebraic curves in Newton's writings is discussed in § 1: the apparatus by which Newton forms the curves, Newton's theorems on forming unicursal curves, his theory of conics, and his theory of (m, n) correspondence. Special Cremona plane and space transformations obtained by Newton's organic method are dealt with in § 2. The article ends with § 3, which shows two different directions in the theory of the constructive formation of plane algebraic curves in the XVIII-XIXth centuries. A synopsis is appended.

The effects are considered of base circle eccentricity, pitch error, dropped tooth, tooth bending under load, and of involute error on the transmission errors of gears. Measurements accurate to ± 1/4 μm show that adjacent pitch errors are frequently randomly distributed. When subject to Fourier analysis, they can give rise to unexpected harmonics both in the errors of a gear and the noise produced by it. A cause of ‘phantom’ gear noise is suggested.

Let H be a real or complex Hilbert space and let L p(1≦p

The classical description of the critical point leads to infinities in such properties as the compressibility and the heat capacity at constant pressure, Cp, as the critical point is approached along a path in the homogeneous fluid. However, this description does not require a singularity in the heat capacity at constant volume, Cv, and supposes that the Helmholtz free-energy, A, is everywhere an analytic function of volume, V, and temperature, T. It is now clear that this description is inadequate, that Cv is infinite at the critical point and that A is a non-analytic function of V and T. The classical description must, therefore, be abandoned and it is shown that we have, in its place, a set of inequalities between the degrees of the singularities in such thermodynamic functions as Cv, compressibility, coefficient of thermal expansion along the orthobaric curve, and the curvature of the vapour pressure line. These inequalities provide powerful tests of even the best modern measurements and enable us to give a fairly complete thermodynamic description of the singularities of the critical region. The transport properties are not so well understood. It is probable that the shear viscosity is non-singular whilst the thermal conductivity is known to be highly singular. This last singularity is probably related to the infinity in Cv. The dimensionless functions—Rayleigh number, Prandtl number, and Grashof number—all become infinite at the critical point.

Following a specific model for the behaviour of a centrifugal pump under advanced cavitation conditions a parameter is derived for the correlation of pump performance when handling different liquids. The parameter's development is based on bubble growth theory, assuming that for the cavitation condition being investigated the bubble radius is much greater than that of the initial nucleus. Published data for a 3 per cent loss in pump head are then used to evaluate the parameter, and to compare it with the ‘thermal parameter’ and that suggested by Barenboim. Whilst all three parameters are reasonably successful with Salemann's data, all fail with Spraker's data. The author then argues that since the data are for limited cavitation conditions a strong Reynolds number influence can be expected, and it is shown that the product of this number (with an empirical fractional exponent) and the derived parameter results in good correlation for each pump. It is expected that the magnitude of the exponent will decrease as the degree of cavitation increases.

Bu yazıda küresel yıldız kümesi Messier 5 deki değişen yıldızlar tetk ik edilm iştir, 'K(I); A(2); A(3) ve A(4) ün nüm erik değerleri tablo halin ­ de g österilm iştir.Şekilde A (2) değerleri m ütekabil log P değerlerine göre izdüşürülmüş* tü r. N oktaların dağılım ında

The streamline curvature technique for the numerical solution of the equations of motion for two-dimensional flow has been analysed. By considering a simple flow case an optimum damping factor for the quasi-orthogonal method was derived as a function of grid aspect ratio, Mach number, and differentiation formula, and the number of iterations required for a given accuracy was found. The stability and accuracy properties of numerical methods for finding the second derivative of a curve were also investigated, and it was found that, in this context, finite difference methods using polynomials were the best and spline methods the worst.

We are grateful to Professor Hoyle for drawing attention to the incorrect sign that we arrived at for the factor λ in our paper (1959), but the changes consequent upon setting this right appear to be quite minor, and if anything lead to more acceptable though not very different, values from those we obtained for the various quantities involved. If the modified Maxwell equations are written now F v μ:v = 4 πJ μ - λK μ , when equations (52) on p. 324 of our paper (1959) should be replaced by E 00 = λ /8 π K 2 0 , E 11 = E 22 = E 33 = λ /8 π e 2 t/T K 2 0 . In mixed form, these are equivalent to E 0 0 = λ /8 π K 0 2 , E 1 1 = E 2 2 = E 3 3 = λ /8 π K 0 2 .

It is shown that the method of Chernoff developed in the preceding paper can be modified to prove the essential self-adjointness on C 0 ∞ ( R m ) of all positive powers of the Schrodinger operator T = − Δ + q if q real and in C ∞ ( R m ) and if T ⩾ −a − b ¦ x ¦ 2 on C 0 ∞ (R m ) .

In [5] S. Holm proposed teststatistics for testing simple hypotheses by means of the probability paper for distribution functions (d.f.) of the form F 0(x) = Φ[(x - μ0)/σ0], where μ0 is location parameter, σ0 scale parameter, and Φ is an absolutely continuous distribution function with Φ(0) = 1/2. If μ0 and (σ0 are known, the hypothesis H 0 is: H 0: H(x) = F 0(x) = Φ[(x−μ0)/σ0], while the three possible alternatives are H 1: H(x) > F 0(x) H 2: H(x) < F 0(x) H 3: H(x) ≠ F 0(x).

Within the class of offspring distributions with given meanm>1 and probability of no offspringp o, the probabilityq of ultimate extinction in a Galton-Watson branching process starting from one individual satisfiesp 0

The exact solutions given in [1] are generalized to the case of cylindrical and spherical sectors rotating about the azimuth relative to the coordinate origin either at a uniform rate or with uniform acceleration (or deceleration). The class of equations of motion of the boundaries of the half-space (in the Cartesian coordinates) which lead to exact solutions of the Fourier equation defined in these domains, is enlarged.

Publisher Summary Thorn [1] has .described seven differentiable catastrophes, and these have been elaborated by Zeeman [2] and by both Thorn and Zeeman in this symposium on dynamic systems. In this paper, we present a beginning of the theory of polyhedral catastrophes and, in particular, we examine analogues of each of the four basic differentiable catastrophes which map the line to the line. The unfoldings of these polyhedral singularities are topologically equivalent to those of the differentiable category, so this alternative view of catastrophe theory can give some additional insight into the structure of differentiable unfoldings. The main advantage of the polyhedral theory is that there is a precise algorithm for determining the singular behavior as opposed to the progressively complicated algebraic calculations required in the differentiable theory.

Publisher Summary This chapter discusses the differential equations associated with Riemannian metrics, which have some invariance properties under conformai transformations. It discusses a possible Riemannian metric, which is to play the role of the Hilbert metric, that is, it is invariant under projective transformations and, in the case of the ball |x| < 1, reduces to hyperbolic Riemannian metric. It is defined in terms of a solution of the boundary value problem for a positive, strictly concave function u defined in Ω.

Associated with Poincare’s “geometric theorem”, there is a number of proven and unproven propositions on fixed points of symplectic (or, as called by Poincare canonical) diffeomorphisms of symplectic manifolds more general than a circular annulus (or in an even more generally, on intersections of so-called Lagrangian submanifolds of a symplectic manifold).

Publisher Summary This chapter focuses on two types of piecewise quadratic spaces and their order of accuracy for Poisson's equation. Error estimates are described, and the results of several numerical experiments are discussed. The results for the vertex method in the chapter, which are exact only for linear polynomials provide an example of the consequence of using a quadrature scheme of insufficient accuracy in relation to the finite element space being used.

Of the variable system s M3, fourty are analysed. The re- sults are given in th e table and are graphically represented in the figüre.