{"title":"On isoptic families of curves","authors":"H. Richmond","doi":"10.1017/S0950184300002743","DOIUrl":"https://doi.org/10.1017/S0950184300002743","url":null,"abstract":"1. Imagine that a number of straight lines, coplanar and concurrent, are united so as to form as it were a rigid frame. Imagine that thiB frame is moved (continuously) in the plane in such a way that two selected lines always touch two cycloids traced in the plane. Then it will be found that Every line of the frame will move so as to envelope a cycloid. Isoptic (and orthoptic) are names used by Charles Taylor for loci on which two tangents of curves intersect at a constant angle. The cycloids form a family of curves in which each two members have the same isoptic locus; they may therefore be described as forming an isoptic family. Isoptic loci are of no great importance or interest. Our aim here is to investigate this and other instances in which curves of a uniform type are enveloped by the various lines of a rigid frame.","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"12 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":"114185687","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}
{"title":"George David Birkhoff","authors":"I. M. H. Etherington","doi":"10.1017/S0950184300002755","DOIUrl":"https://doi.org/10.1017/S0950184300002755","url":null,"abstract":"","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"38 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":"121432198","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}
{"title":"Inequalities for solutions of linear differential equations a contribution to the theory of servomechanisms","authors":"Hans Bückner","doi":"10.1017/S0950184300002986","DOIUrl":"https://doi.org/10.1017/S0950184300002986","url":null,"abstract":"Consider the n th order differential equation where the coefficients c v are real constants and f is a real function continuous in the interval a ≦ x ≦ b . The following theorem will be proved in §4: If the characteristic equation of (I) has no purely imaginary roots, then a particular integral η ( x ) can always be found which satisfies the inequality where C is a certain function of the c v only and M is the maximum of |f|. In particular we may take C = 1 if all roots of the characteristic equation are real.","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"15 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":"126733002","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}
{"title":"On the Newton-Raphson method of Approximation","authors":"H. Richmond","doi":"10.1017/S0950184300000082","DOIUrl":"https://doi.org/10.1017/S0950184300000082","url":null,"abstract":"simple proof of the theorem that a skew symmetric determinant of even order is a perfect square. This can be done by examination of the way in which cycles of two indices may be agglomerated in cycles of more indices, but it would hardly seem to be so simple as Sylvester believed. One can, however, easily enumerate the terms in the square root, the Pfaffian. For the squared terms in the skew determinant correspond exclusively to permutations containing cycles of two indices only, since (if) connotes — ay ciji, or a?.. Thus we have to find in how many ways 2m indices may be put into m such cycles. For first cycle take 1 and any a from the remaining 2m 1 indices; for second cycle take the next surviving index in natural order and any b from the remaining 2m 3; and so proceed. The number of terms in the Pfaffian is thus (2m 1)(2m 3)(2m 5) ... 5.3.1, a factorial composed of odd numbers. This is a well-known result.","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"23 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":"126814213","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}
{"title":"On pedal tetrahedra","authors":"R. Robinson","doi":"10.1017/S0950184300002561","DOIUrl":"https://doi.org/10.1017/S0950184300002561","url":null,"abstract":"P 1 , = -o (rq — sp), p which gives the result required. Since my Solution 2 was, quite unintentionally, rather unfair to the method of partial derivatives, I feel that I ought to draw attention to this shorter solution. The fact that the above solution is merely shorter than the one which I gave does not however detract from the practical advantages of the differential method. Any experienced teacher knows that the step which presents real difficulty to the beginner is the obtaining of equation (1) above. Although in the case of the example which I happened to choose for illustration (and it may not have been the best for the purpose) the above solution by partial derivatives happens to be quite as short as the solution by differentials, the fact remains that, while the technique of differentiation, when once understood, is almost \" fool-proof,\" the pitfalls for the beginner in the solution given above are well known to every teacher of the subject. While the solution of a problem by partial derivatives may be quite a difficult piece of manipulation, exactly the same technique is required for the solution of a problem by differentials, however simple or complicated the problem in question may happen to be.","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"17 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":"126828205","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}
{"title":"Dirichlet's Integrals","authors":"L. Mordell","doi":"10.1017/S0950184300000124","DOIUrl":"https://doi.org/10.1017/S0950184300000124","url":null,"abstract":"which is exactly what it would be using the theorem. But the theorem is true for n = 1 and n = 2, so it is true for n = 3, and hence similarly for all positive integral values of n. Applications, (i) Expansion for 2 cos n 6 in terms of 2 cos 9. Let x = cos 6 + i sin 9, y = cos 9 — i sin 9; then x + y = 2 cos 0 and xy = 1. Also x = cos n 0 + i sin w # ; y — cos n 9 — i sin n 0 so that £\" + y = 2 cos n 9. Substituting these in the theorem, we obtain","PeriodicalId":417997,"journal":{"name":"Edinburgh Mathematical Notes","volume":"26 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":"129449159","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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