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The paper deals with state estimation of nonlinear stochastic dynamic systems. In particular, the iterated extended Kalman filter is studied. Three recently proposed iterated extended Kalman filter algorithms are analyzed in terms of their performance and specification of a user design parameter, more specifically the step-length size. The performance is compared using the root mean square error evaluating the state estimate and the noncredibility index assessing covariance matrix of the estimate error. The performance and influence of the design parameter, are analyzed in a numerical simulation.

The Fourier method is a rather effective and very simple regularization method for solving some ill-posed problems. The known works on this method are all limited to the a priori choice of the regularization parameter. In this paper, we will systematically consider the a posteriori choice of the regularization parameter, and the corresponding error estimates between the exact solution and its approximation are given. Numerical examples show the effectiveness of the a posteriori method, and the comparisons of the numerical effect between the a posteriori and the a priori Fourier methods are also taken into account.

The condition for the correction of coma in a centred optical system is the well-known Sine condition. This has been proved by Clausius, Helmholtz, Hockin and others, and the importance of this condition in the design of optical systems has been pointed out by Abbe, Steinheil, Conrady, and others. So far as the Author is aware no discussion of the effects of failure to satisfy this condition has been published, and the present paper shows how to obtain the relation between the Coma of a system and the errors in the Sine Condition.

This paper deals with a determination of the effective resistance and inductance of a helical coil of great length, composed of thin wire, wound on a cylinder whose radius is large in comparison with that of the wire. The pitch of the winding is not small, so that the problem cannot be treated by the method of Cohen. The method employed depends upon the use of a type of "helical coordinates" defining the position of any point, and of the general theorem relating to orthogonal systems of coordinates. A solution is obtained for the internal and external forces, corresponding to a given impressed electro-motive force, in the form of a Fourier series of which only the initial terms require calculation. The value of the effective current across any section is obtained, and thence the inductance and resistance. The results are of simple character, and are expressed in general in terms of the ber and bei functions of Lord Kelvin, whose results for a straight wire appear as a limiting case. Certain particular cases are worked out in detail and formulae obtained in terms of elementary functions which are suitable for a very high or a low frequency. Their necessary limitations are also examined numerically. For a high frequency, it is found that the change of self-inductance due to twisting of the wire tends to vanish, and that the change of resistance tends towards a value independent of the frequency.

In circuits possessing constant inductance, resistance and capacity the differential equations for the currents and the voltages are linear with constant co-efficients, and may therefore be solved by aid of the known simple properties of symbolic operators. The symbolic operator method proves to be very compendious in problems concerning the determination of the primary and secondary currents and voltages of transformers whenever the applied E.M.F. can be expressed as an exponential function of the time, or when it consists of a sudden application of a constant or an exponential function, and also when it is impulsive. In the Paper the method is first applied to a pair of indirectly coupled circuits - i.e., circuits that are insulated from each other. The cases of pure magnetic coupling, pure electric coupling and of mixed magnetic and electric coupling are treated in turn and shown to be operationally all of the same form. Then general operational equations are obtained for direct coupling and for mixed direct and indirect coupling - this latter, of course, includes all types of autotransformers, both high and low frequency. Here again the operational equations are of the same form as in the case of magnetic coupling. The equations readily show that if the primary applied voltage be a pure sine or cosine alternating E.M.F. the well-known circle diagram for determining the instantaneous magnitudes and phases of the primary and secondary currents in an ordinary transformer is applicable to the most general type of transformer, and is capable of certain extensions. It is then shown that a similar circle diagram can be used for obtaining the primary and secondary currents in any general transformer under the application of the quasi-steady conditions of a slowly-damped alternating voltage. A further new theorem arises here, which affirms that a similar graphical method of computation can be used for deducing the voltages at the terminals of the primary and secondary condensers in a high-frequency transformer under steady or quasi-steady alternating current. In the second section of the Paper it is shown how to calculate accurately the free periods and the damping factors of a coupled system with two degrees of freedom, and approximate formulae are also deduced. The third section passes to the investigation of the currents and voltages produced in the circuits by sudden application or removal of a constant, a harmonic, or a damped harmonic E.M.F., and of those produced by the application of electromotive impulses. For this purpose formulae expressing the effect of the various types of symbolic operator occuring in the Paper, on impulsive E.M.F.s and on sudden applications, are deduced and tabulated. Using these tables of integrals it is very easy to write down quickly the solutions appropriate to any given transformer under the action of any of the types of E.M.F. specified. These tables include all the cases likely to arise in wireless telegraphy and in ordinary transformers. As an example of the application of the methods the problem of a primary spark-circuit and its antenna is worked out under resonance conditions, and for various couplings and time graphs are made of the primary and secondary currents and voltages. A further example is the solution for the incidence of a natural electric wave (a stray) on a receiving antenna and its resonant secondary.

Methods are given for the rapid calculation of the mutual inductance of two coaxial circular currents. Maxwell's first formula is converted into theta-functions, and then expanded in a Jacobian q series. The logarithmic values of this series for various values of q have been tabulated in a previous Paper by the author. When the circles are near one another a series for M is given in terms of q1, where q1 is the complement of q. In this Paper the author treats Maxwell's second formula in a similar way. A table of the values of these series found, computed to six decimal figures by T. Tishima, is given. The chief advantages of this table are that nearly all practical cases are included within a short range of the argument, and the calculation is simple, as the numbers in the difference columns are small. By the help of these tables and series the mutual inductance between two coaxial currents can be easily computed to a high degree of accuracy.

Prof. Fleming exhibited an apparatus for the production of vibrations on strings loaded or unloaded. This method is an improvement on that of Melde, in which transverse vibrations are created on a string attached to the prong of a tuning fork. Melde's method has the disadvantage that it is not possible to put great tension on the string without stopping the fork, or to use very heavy or loaded strings without using forks of great size. Moreover, the frequency cannot be varied except by changing the fork. In the Author's method the vibrations are produced on a string by attaching one end to the shaft of a small continuous-current motor of about 1/8 H.P. The motor has on one end of its shaft a mechanism for counting revolutions, and on the other end a disc which has a crank pin inserted in its outer face, and to this pin is attached a light crankshaft connected at its outer extremity with a rocking lever. The string is fastened to a hook on the crankshaft near to the crank pin. The other end of the string is attached to a fixed point which can be moved by means of a screw, in some cases a spring balance being interposed to measure the tension. When the motor is started the string has a circular motion given to its end which is equivalent to two simple harmonic motions at right angles to each other. If the tension is rightly adjusted the string then vibrates in sections, and the number of sections can be adjusted by altering the tension. The distance from node to node can then easily be measured and the frequency determined from the speed of the motor. In this way the velocity of the wave is measured, and can be compared with the velocity determined by taking the square root of the quotient of the tension by the linear density of the string. A number of tables were exhibited showing the velocity of the wave motion on strings of various kinds confirming the known laws of string vibrations. This method is particularly useful in studying the properties of loaded strings. Strings made of flexible cotton cord can be loaded with glass or wooden beads and set in vibration. In this manner it can be shown experimentally that when the wave length on the string extends over a distance of more than 8 or 10 loads, the string vibrates as if the loading matter were uniformly distributed, but the string cannot propagate vibrations when the half wave length approaches equality to the distance between two loads. It is also possible to show by this apparatus very prettily the reflection of a wave at a load placed at any point on the string, and also that this reflection is reduced by tapering off the loading. We can thus imitate with this loaded vibrating string all the phenomena of inductive loading in telephone cables on the Pupin system. The theory of the vibrations of a loaded string has been studied by many mathematicians - for example, by Lagrange, Lord Rayleigh, Prof. P. G. Tait, and a very interesting Paper was published on the subject by Mr. Charles Godfrey in the "Philosophical Magazine" for April, 1898. This method of causing a string to vibrate by a motor affords a simple method of demonstrating all the laws of vibrating strings to a class

A method of drawing the cusped wave-fronts produced by refraction and reflection at a spherical surface and a graphic treatment of the elementary theory of the rainbow are described in the Paper. The method is based upon the properties of the centre of junction. In the case of reflection at a spherical surface in which aplanatic points are not available the position of the junction-centre is obtained by elementary geometry. By the use of junction-centres the distribution of the successive foci obtained on refraction and reflection at a spherical surface is readily plotted. If in the case of a rainbow it is assumed that the emergent portion of the effective pencil, as well as the incident portion, is one of parallel rays, then the distribution of foci with regard to the drop is a symmetrical one. The caustics on refraction and reflection are also readily drawn as loci of points and envelopes of rays. Hence the wave-surfaces can be obtained. These in some regions are cusped and the caustics are the loci of the cusps. In the case of pencils that are effective in producing rainbows, the lengths of the incident portion of the chief ray intercepted by the spherical drop of radius, r, and a concentric sphere of radius, r. μ, are in the ratio 1 to n + 1, where n is the number of internal reflections. This leads to a geometrical construction for finding the points of incidence of the chief rays of the effective pencils. These rays are at minimum deviation, and the same chief ray remains at minimum deviation although the associated pencil may be converging or diverging.