Pub Date : 1900-01-01DOI: 10.1017/S0950184300000161
A. Macintyre, W. Rogosinski
{"title":"Some elementary inequalities in function theory","authors":"A. Macintyre, W. Rogosinski","doi":"10.1017/S0950184300000161","DOIUrl":"https://doi.org/10.1017/S0950184300000161","url":null,"abstract":"","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127657016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.1017/S0950184300002731
A. G. Walker
1. The following description of the projective geometry of a finite number of points in 2-space is almost certainly known to those acquainted with projective geometry or with modern algebra. The object of this brief account is to show how certain finite systems can be presented in a form easily understood by students, and how they provide simple but instructive examples of fundamental ideas and “constructions.” The fact that these examples belong to a geometry which is essentially non-Euclidean has great teaching value to those students who are apt to confuse projective geometry with the “method of projection” in Euclidean geometry. The underlying algebra is described briefly in § 4, but an understanding of this is not essential to the geometry. This algebraic work may, however, be of interest to those to whom Galois fields are fairly new.
{"title":"Finite protective geometry","authors":"A. G. Walker","doi":"10.1017/S0950184300002731","DOIUrl":"https://doi.org/10.1017/S0950184300002731","url":null,"abstract":"1. The following description of the projective geometry of a finite number of points in 2-space is almost certainly known to those acquainted with projective geometry or with modern algebra. The object of this brief account is to show how certain finite systems can be presented in a form easily understood by students, and how they provide simple but instructive examples of fundamental ideas and “constructions.” The fact that these examples belong to a geometry which is essentially non-Euclidean has great teaching value to those students who are apt to confuse projective geometry with the “method of projection” in Euclidean geometry. The underlying algebra is described briefly in § 4, but an understanding of this is not essential to the geometry. This algebraic work may, however, be of interest to those to whom Galois fields are fairly new.","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"109 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124573776","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.1017/S095018430000272X
W. Rogosinski
1. It is now nearly half a century since H. Lebesgue, whose obituary the reader may have seen in Nature not so long ago, created his theory of the integral which since then has superseded in modern analysis the classical conception due to B. Riemann. It is, I think, regrettable that knowledge of the Lebesgue integral seems to be still largely confined to the research worker. There is nothing unduly abstract or unnatural in this theory, nor anything in the proofs which would be too difficult for a good honours student to grasp. If the aim of university education be the teaching of general ideas and methods rather than that of technicalities, then the modern notion of the integral should not be omitted from the mathematical syllabus. It is the purpose of this purely expository note to sketch the build up of both the Riemann and the Lebesgue integral on the common geometrical basis of “measure” and thus to make evident to the uninitiated reader the striking advantages of the new integral.
{"title":"Measure and Integral","authors":"W. Rogosinski","doi":"10.1017/S095018430000272X","DOIUrl":"https://doi.org/10.1017/S095018430000272X","url":null,"abstract":"1. It is now nearly half a century since H. Lebesgue, whose obituary the reader may have seen in Nature not so long ago, created his theory of the integral which since then has superseded in modern analysis the classical conception due to B. Riemann. It is, I think, regrettable that knowledge of the Lebesgue integral seems to be still largely confined to the research worker. There is nothing unduly abstract or unnatural in this theory, nor anything in the proofs which would be too difficult for a good honours student to grasp. If the aim of university education be the teaching of general ideas and methods rather than that of technicalities, then the modern notion of the integral should not be omitted from the mathematical syllabus. It is the purpose of this purely expository note to sketch the build up of both the Riemann and the Lebesgue integral on the common geometrical basis of “measure” and thus to make evident to the uninitiated reader the striking advantages of the new integral.","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114975110","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.1017/S0950184300000173
D. Martin
If Z is a fixed point on the surface of the earth (assumed spherical) and P is the North Pole, then the locus of a point X, which moves in such a way that the angle a between the great circle arcs PX, ZX is constant, is called a curve of constant bearing. a is measured clockwise from XP; it is then the great circle bearing of Z from X as defined in navigation, and lies within the range 0° to 360°. Curves of constant bearing are of some importance in navigation because, if a ship or aircraft at X takes a bearing of a radio station at Z, the position line so obtained is an arc of such a curve. Nevertheless, few properties of the curves seem to be recorded; the reason is probably that practical navigators are interested not in the actual curves in their entirety but in the projections on a Mercator chart of comparatively short lengths of them. In this note some simple properties of the curves are obtained; the derivation of the results is very straightforward and, needless to say, no originality is claimed. We begin by writing down the equation of a curve of constant bearing. Let the latitudes of X and Z be , 0 respectively, and let the meridian of Z be that of zero longitude; the longitude A of X is considered as positive or negative according as it is Easterly or Westerly. Since, by convention, the angles of a spherical triangle cannot exceed 180°, two cases a < 180° (fig (i)) and a > 180° (fig. (ii)) must be considered. Then, in both cases, by the Four Part Formula of Spherical Trigonometry, the equation of the curve of constant bearing is
{"title":"Some properties of the curve of constant bearing","authors":"D. Martin","doi":"10.1017/S0950184300000173","DOIUrl":"https://doi.org/10.1017/S0950184300000173","url":null,"abstract":"If Z is a fixed point on the surface of the earth (assumed spherical) and P is the North Pole, then the locus of a point X, which moves in such a way that the angle a between the great circle arcs PX, ZX is constant, is called a curve of constant bearing. a is measured clockwise from XP; it is then the great circle bearing of Z from X as defined in navigation, and lies within the range 0° to 360°. Curves of constant bearing are of some importance in navigation because, if a ship or aircraft at X takes a bearing of a radio station at Z, the position line so obtained is an arc of such a curve. Nevertheless, few properties of the curves seem to be recorded; the reason is probably that practical navigators are interested not in the actual curves in their entirety but in the projections on a Mercator chart of comparatively short lengths of them. In this note some simple properties of the curves are obtained; the derivation of the results is very straightforward and, needless to say, no originality is claimed. We begin by writing down the equation of a curve of constant bearing. Let the latitudes of X and Z be </>, <f>0 respectively, and let the meridian of Z be that of zero longitude; the longitude A of X is considered as positive or negative according as it is Easterly or Westerly. Since, by convention, the angles of a spherical triangle cannot exceed 180°, two cases a < 180° (fig (i)) and a > 180° (fig. (ii)) must be considered. Then, in both cases, by the Four Part Formula of Spherical Trigonometry, the equation of the curve of constant bearing is","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127064151","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.1017/S0950184300002950
S. Paterson
The series in which r is zero or an integer is rapidly convergent if x is large but may be very slowly convergent if x is small. The object of this note is to derive an alternative series for S 2r ( x ) which is rapidly convergent for small values of x .
{"title":"The summation of a slowly convergent series","authors":"S. Paterson","doi":"10.1017/S0950184300002950","DOIUrl":"https://doi.org/10.1017/S0950184300002950","url":null,"abstract":"The series in which r is zero or an integer is rapidly convergent if x is large but may be very slowly convergent if x is small. The object of this note is to derive an alternative series for S 2r ( x ) which is rapidly convergent for small values of x .","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132067635","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.1017/S0950184300002676
B. B. Smith
{"title":"Note on an alternant suggested by statistical theory","authors":"B. B. Smith","doi":"10.1017/S0950184300002676","DOIUrl":"https://doi.org/10.1017/S0950184300002676","url":null,"abstract":"","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"122 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130500221","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.1017/S0950184300002962
I. Good
The purpose of this note is to generalise the Dirichlet-Liouville formula which expresses a certain type of multiple integral in terms of a single integral. In our formula the multiple integral will involve several arbitrary functions instead of only one, and it will be expressed as a product of single integrals.
{"title":"A Generalisation of Dirichlct's Multiple Integral","authors":"I. Good","doi":"10.1017/S0950184300002962","DOIUrl":"https://doi.org/10.1017/S0950184300002962","url":null,"abstract":"The purpose of this note is to generalise the Dirichlet-Liouville formula which expresses a certain type of multiple integral in terms of a single integral. In our formula the multiple integral will involve several arbitrary functions instead of only one, and it will be expressed as a product of single integrals.","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126596617","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.1017/S0950184300000148
L. M. Brown
/ / P l t P2, P3, P* are four points on a circle C, and P234 is the orthocentre of triangle P2 P 3 P4. Piu the orthocentre of triangle P1P3 P4 and so on, then the quadrilateral P234 PISA -P124 -P123 t s congruent to the quadrilateral PlP2PsPi. This theorem seems to be due to Steiner (Oes. Werke, 1, p. 128; see H. F. Baker, Introduction to Plane Geometry, 1943, p. 332) and has appeared frequently since in collections of riders on the elementary circle theorems. It is clear that P234 P134 P124 P123 lie on a circle C1234 equal to the original circle C. But also angle P 3 P134 P4 = P4 Pj P 3 = P4 P2 P 3 = PsP23iPi (with angles directed and equations modulo 77), and hence P 3 P4 P134 P234 lie on a circle C3i equal to C, and which is in fact the mirror image of C in P3P4. Similarly we obtain circles C12, C13, C14, C23, C24, so that we have in all eight circles with four points on each. If any one of these be taken as the original circle, the same system of eight circles is obtained ; if, e.g., we begin with PsPiP^ P234 on the circle C34, the four orthocentres are Pl 5 P2, P123, P m lying on C]2 and the remaining circles are the images of C34 in the six sides of the quadrangle P3 P4 P134 P234. Call this configuration KA. Let us now take a fifth point P5 on C. Then any four of Pj P2 P3 P4 P5 give a Ki. We have in fact five points Pa . . . . P5, ten points P123 . . . . P345, a circle C, ten circles C12 . . . . C45 and five circles C1234 . . . . C2345. Then the circles Cl2U C1235 C12i5 C13ii C2345 all pass through a point Pi2u&, completing a system of 16 poinis and 16 circles, five points on each circle and five circles through each point. We may show this by taking the circle C12, e.g., on which lie the five points P 1 P 2 P ) 2 3 Pi 2 4 Pi 2 5 and build up the K^s obtained by taking these four at a time. Use a parallel notation and write Q1 = Plt Q2 = P2,
{"title":"On a Chain of Circle Theorems","authors":"L. M. Brown","doi":"10.1017/S0950184300000148","DOIUrl":"https://doi.org/10.1017/S0950184300000148","url":null,"abstract":"/ / P l t P2, P3, P* are four points on a circle C, and P234 is the orthocentre of triangle P2 P 3 P4. Piu the orthocentre of triangle P1P3 P4 and so on, then the quadrilateral P234 PISA -P124 -P123 t s congruent to the quadrilateral PlP2PsPi. This theorem seems to be due to Steiner (Oes. Werke, 1, p. 128; see H. F. Baker, Introduction to Plane Geometry, 1943, p. 332) and has appeared frequently since in collections of riders on the elementary circle theorems. It is clear that P234 P134 P124 P123 lie on a circle C1234 equal to the original circle C. But also angle P 3 P134 P4 = P4 Pj P 3 = P4 P2 P 3 = PsP23iPi (with angles directed and equations modulo 77), and hence P 3 P4 P134 P234 lie on a circle C3i equal to C, and which is in fact the mirror image of C in P3P4. Similarly we obtain circles C12, C13, C14, C23, C24, so that we have in all eight circles with four points on each. If any one of these be taken as the original circle, the same system of eight circles is obtained ; if, e.g., we begin with PsPiP^ P234 on the circle C34, the four orthocentres are Pl 5 P2, P123, P m lying on C]2 and the remaining circles are the images of C34 in the six sides of the quadrangle P3 P4 P134 P234. Call this configuration KA. Let us now take a fifth point P5 on C. Then any four of Pj P2 P3 P4 P5 give a Ki. We have in fact five points Pa . . . . P5, ten points P123 . . . . P345, a circle C, ten circles C12 . . . . C45 and five circles C1234 . . . . C2345. Then the circles Cl2U C1235 C12i5 C13ii C2345 all pass through a point Pi2u&, completing a system of 16 poinis and 16 circles, five points on each circle and five circles through each point. We may show this by taking the circle C12, e.g., on which lie the five points P 1 P 2 P ) 2 3 Pi 2 4 Pi 2 5 and build up the K^s obtained by taking these four at a time. Use a parallel notation and write Q1 = Plt Q2 = P2,","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122244131","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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