The life and work of Prof. George Chrystal (1851-1911)

This thesis is principally concerned with George Chrystal's life and his work, mainly in three directions viz., as an experimentalist, a mathematician, and an educationist. The main object is to bring to light the work of a personality who is representative of many more who are always forgotten. The majority of historians of science consider the works of the giants in science, ignoring totally the contributions made by the less prominent people like Prof. George Chrystal. In fact their contributions serve as one of the most important factors in propagation of scientific knowledge. His main contributions: verification of Ohm's Law experimentally; Non-Euclidean geometry; differential equations; text books on algebra; theory of seiches; institution of leaving certificate examination in Scottish education and many more have been discussed in detail. A survey of Chrystal's general thought is given in so far as it may be gathered from his scattered remarks. The references are mentioned by numerals in the superscript, details of which are given at the end of each chapter. The main text consists of six chapters. There are three appendices at the end,' Appendix 'A' consists of his correspondence with different scientists, most of which is still unpublished. Appendix 'B' contains a bibliography of his contributions in chronological order, and Appendix 'C contains his three Promoter's addresses. Tables and figures are attached at their proper places, including some rarely available photographs

On the orderly listing of permutations

D.H. Lehmer states that by an orderly listing of permutations is meant a generation for which it is possible to obtain the k th permutation directly from the number k, and conversely, given a permutation, it is possible to determine at once its rank, or serial number, in the list without generating any others. In the following discussion several methods of obtaining an orderly listing are considered, especially with respect to the recovery of information regarding the number of inversions in a given permutation

A variable-channel queuing model with a limited number of channels

M.S.Joseph J. Mode

I. Circumspheres in Hilbert space, II. Automatic handling of finite-dimensional, nonassociative algebras

Part I. A nonempty, bounded subset X of a Hilbert space has a unique circumsphere S(X). Its center, c(X), belongs to the closure of the convex hull of X. Its radius r(X) never exceeds d/SQRT.(2) where d is the diameter of X. There is a closed bounded set X for which c(X) does not belong to the convex hull of X, X (INTERSECT) S(X) is empty, and r(X) = d/SQRT.(2);A set X is nonredundant if x lies outside S(X-x) for all x (ELEM) X. The closure of any bounded set in R(\u27n) contains a nonredundant subset Y with S(Y) = S(X). Also. (VBAR)Y(VBAR) (LESSTHEQ) n + 1, and c(Y) is the unique point in the convex hull of Y which is equidistant from each point of Y. This characterization leads to a new algorithm for finding circumspheres in R(\u27n);Part II. A multilinear function defined on a nonassociative algebra X is completely determined by its table, the matrix of a linear transformation from an appropriately defined vector space into X. Operations on tables are defined corresponding to operations on multi-linear functions. This method was used to find identities in the free alternative algebra on three generators. Let A, B, and C be the generators and X the associator (A,B,C). The following identities along with their permuted images have a linear span of dimension 16 which contains all identities in which each generator occurs exactly twice. (1) - ((AB)X)C + ((BA)X)C + (A,B,X)C; (2) - ((XA)B)C + ((XB)A)C + (A,B,X)C; (3) ((AB)C)X - ((BA)C)X - C(A,B,X) - 2XX; (4) - ((AB)C)X + ((AC)B)X + ((CA)B)X - ((CB)A)X - (A,B,X)C + (B,C,X)A; (5) ((AX)B)C - ((AX)C)B - ((CX)A)B + ((CX)B)A + (A,B,X)C - (B,C,X)A; (6) ((CA)B)X - ((CB)A)X - ((CX)A)B + ((CX)B)A; (7) ((CX)A)B - ((CX)B)A + ((XC)A)B - ((XC)B)A - (A,B,X)C - C(A,B,X) - 2XX; (8) ((XB)A)C - ((XB)C)A + ((XC)A)B - ((XC)B)A - B(A,C,X) - C(A,B,X); (9) 3((XA)B)C - 3((XA)C)B - (A,B,X)C + (A,C,X)B - (B,C,X)A - 2A(B,C,X) - B(A,C,X) + C(A,B,X). As a byproduct, we generated an example of an alternative algebra of dimension 307

Tidal phenomena at inland boreholes near Cradock

Observations begun in 1905 and carried on at intervals until the present year on a group of wells on a farm at Tarka Bridge, Cradock District, are described in detail.The wells have not been bored very deep, the deepest being 225 feet, but it is obvious that the bores connect with deeply extending fissures, as the waters issue at temperatures of about 80° accompanied by large quantities of natural inflammable gas (methane), while sulphuretted hydrogen is present in notable quantities in solution in the water. The wells are 2,700 feet above sea-level and over 100 miles from the coast.Measurements of the pressure at which the water issues show a remarkable fluctuation, in some respects analogous to the tidal fluctuations of the sea.A series of direct measurements covering several days established the fact that there was a real fluctuation both in the amount of water discharged and in the well- pressure. Continuous records were then obtained over longer periods by means of clock -driven, self -recording apparatus in order to study the precise nature of the fluctuations. The longest continuous record obtained extends over a. period of fifteen weeks. This graphical record shows that the semi-diurnal fluctuations attain a maximum amplitude at fortnightly intervals at times corresponding to the times of new moon and full moon throughout the fifteen-weeks' period.This record further demonstrates the fact that the mean daily water pressure rises with each fall of barometric pressure, and falls with each rise in barometric pressure as recorded concurrently at the farm by means of a barograph instrument. The time scale on this fifteen -week record is about 11 inches per week.Records obtained for shorter periods on a time scale of 14 inches per day were found to be much more suited for detailed critical examination and analysis.. In particular, the record for a certain fortnight during . which the barometric pressure was very steady (and consequently its interfering effect almost negligible) was selected. The times of all the . turning-points were carefully determined in terms of South African official time. The heights of all the turning-points of the curve were also determined in inches.Similarly, the co-ordinates of all the turning-points for that fortnight were determined on the tide gauge records of the South African ports of Cape Town, Port Elizabeth, East London, and Durban.The original Tarka Bridge record for the fortnight was then subjected to a process of harmonic analysis for the purpose of determining the periods of the principal harmonic components of the curve. The particular method used was described by Chrystal as the method of " Residuation," (Trans. Roy. Soc., Edinburgh, vol. xlv., part 2, pp. 385 -7). This method is applicable to comparatively short curves and involves no assumptions as to periods or the causes operating to produce the curve. One by one the various simple harmonic components are disentangled from the compound curve with their periods unaltered but with their amplitudes considerably reduced. In this way components were isolated from the Tarka Bridge curve having the following wave periods :- 1. 12 hrs. 27 min. [probably divisible into 12 hrs. 26 min. and 12 hrs. 0 min.] 2. 23 hrs. 57 min. 3.. An anharmonic residuum which appeared to be the vertical inversion of the barograph curve for the fortnight.For general comparison all these measured data were plotted in parallel lines on the same time scale, and the general resemblance of the well curve to the curves of the coastal tide records demonstrated.The original Tarka Bridge record for the fortnight was then subjected to a process of harmonic analysis for the purpose of determining the periods of the principal harmonic components of the curve. The particular method used was described by Chrystal as the method of "Residuation," (Trans. Roy. Soc., Edinburgh, vol. xlv., part 2, pp. 385 -7). This method is applicable to comparatively short curves and involves no assumptions as to periods or the causes operating to produce the curve. One by one the various simple harmonic components are disentangled from the compound curve with their periods unaltered but with their amplitudes considerably reduced.In this way components were isolated from the Tarka Bridge curve having the following wave periods :- 1. 12 hrs. 27 min. [probably divisible into 12 hrs. 26 min. and 12 hrs. 0 min.] 2. 23 hrs. 57 min. 3.. An anharmonic residuum which appeared to be the vertical inversion of the barograph curve for the fortnight.Component No.2 was obviously not a simple harmonic function. It seemed to me to be composed of several harmonics of approximately diurnal period, but on the scale on which the analysis was being conducted the practical limit of the method had been reached. Accordingly no finer dissection was attempted.The above results may be compared with the well-known principal harmonic components of marine tides. 1. Principal Lunar semi - diurnal tide -period 12 hrs. 25 min. 14* sec. 2. Principal Solar semi-diurnal tide -- period 12 hrs. 3. Three diurnal tides with periods - 23 hrs. 56 min.; 24 hrs. 4 min. ; 25 hrs. 40 min. 9.5 sec.GENERAL REMARKS: The foregoing results seem to establish beyond question that the fluctuations in these wells are to be attributed directly or indirectly to extra- terrestrial causes, but the precise nature of the connection is not by any means clear.The wells are situated over 160 miles from the coast, at an altitude of over 2,700 feet above sea-level. High water at Tarka Bridge occurs about 14* hours after high water at East London, while the lag in the case of low water is nearly 15 hours.The principal conceivable theories to account for the phenomena would appear to group themselves in three classes :- A. Theories depending on the direct gravitative influence of the sun and moon on the land or the underground water. B. Theories depending on the action of the marine tides on the coast loading and distorting the land. C. Theories depending on the action of marine tides in periodically reducing the freedom of outflow of underground water through submarine springsNo attempt is at present made to state or discuss these theories. It is felt that a satisfactory theory can be arrived at only by the co-operative discussion of the subject by astronomers, geologists, and hydraulicians.Tidal wells are known in many parts of the world, but practically all are within 3 or 4 miles of the seashore and at no considerable altitude. The records from the Orisino Bore in Australia do not show the periodicity of marine tides.One case is reported at Lille in France, 40 miles from the coast, but at no great height above sea-level. The evidence supporting the tidal claim of this well is not quite satisfactory.It is believed that there is no other record of an inland well showing fluctuations of true lunar periodicity

The Fundamental Principles of Algebra

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The effect of stratification and bathymetry on internal seiche dynamics

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Civil and Environmental Engineering, 2000.Includes bibliographical references.by Paul David Fricker.Ph.D

Faith and criticism in post-disruption Scotland, with particular reference to AB Davidson, William Robertson Smith, and George Adam Smith

Available from British Library Document Supply Centre- DSC:D38325/81 / BLDSC - British Library Document Supply CentreSIGLEGBUnited Kingdo

Mathematics and general education

Democracy and General Education My purpose in Part I is to develop a model of general mathematical education: that is, to identify aims appropriate to a course of mathematical education which forms part of a programme of general education. To do so presumes, of course, that it is possible to justify both the inclusion of mathematics-related aims and content in the curriculum, and their organisation around a unit entitled 'mathematics'. I will offer arguments for both these presuppositions, as well as for my model of general mathematical education