On the Law Which Regulates the Relative Magnitude of the Areas of the Four Orifices of the Heart

Herbert Davies
1869 Proceedings of the Royal Society of London  
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more » ... ntent at http://about.jstor.org/participate--jstor/individuals/early-journal--content. JSTOR is a digital library of academic journals, books, and primary source objects. JSTOR helps people discover, use, and build upon a wide range of content through a powerful research and teaching platform, and preserves this content for future generations. JSTOR is part of ITHAKA, a not--for--profit organization that also includes Ithaka S+R and Portico. For more information about JSTOR, please contact support@jstor.org. 1870.] On the Areas of the four orifices of the Heart. 1870.] On the Areas of the four orifices of the Heart. Trans. vol. clv. p. 653); and as I there, in the case of three variables, introduced a set of three arbitrary constants in order to comprise a group of expressions in a single formula, so here, in the case of four variables, I introduce with the same view two sets of four arbitrary constants. If these constants be represented by a, F, y, , a', F', y', 8', I consider the conic of five-pointic contact of a section of the surface made by the plane w--z'= 0, where w=aox+y+yz+ t, and zw'=-c'x+3'y+y'z+-'t, and k is indeterminate; and then proceed to determine k, and thereby the azimuth of the plane about the line w = 0, zr'=0, so that the contact may be sixpointic. The formulae thence arising turn out to be strictly analogous to those belonging to the case of three variables, except that the arbitrary quantities cannot in general be divided out from the final expression. In fact, it is the presence of these quantities which enables us to determine the position of the plane of section, and the equation whereby this is effected proves to be of the degree 10 in : 't'=-k, and besides this of the degree 12n-27 in the coordinates x, y, z, t (n being the degree of the surface), giving rise to the theorem above stated. Beyond the question of the principal tangents, it has been shown by Clebsch and Salmon that on every surface U a curve may be drawn, at every point of which one of the principal tangents will have a fourpointic contact. And if n be the degree of U, that of the surface S intersecting U in the curve in question will be 1 ln--24. Further, it has been shown that at a finite number of points the contact will be five-pointic. The number of these points has not yet been completely determined; but Clebsch has shown (Crelle, vol. lviii. p. 93) that it does not exceed n(lln-24) (14n-30). Similarly it appears that on every surface a curve may be drawn, at every point of which one of the principal tangent conics has a seven-pointic contact, and that at a finite number of points the contact will become eight-pointic. But into the discussion of these latter problems I do not propose to enter in the present communication. March 17, 1870. Capt. RICHARDS, R.N., Vice-President, in the Chair. The following communications were read: I. "On the Law which regulates the Relative Magnitude of the Areas of the four Orifices of the Heart." I propose in this communication to inquire whether any law can be discovered which determines the relative magnitude of the areas of the Trans. vol. clv. p. 653); and as I there, in the case of three variables, introduced a set of three arbitrary constants in order to comprise a group of expressions in a single formula, so here, in the case of four variables, I introduce with the same view two sets of four arbitrary constants. If these constants be represented by a, F, y, , a', F', y', 8' , I consider the conic of five-pointic contact of a section of the surface made by the plane w--z'= 0, where w=aox+y+yz+ t, and zw'=-c'x+3'y+y'z+-'t, and k is indeterminate; and then proceed to determine k, and thereby the azimuth of the plane about the line w = 0, zr'=0, so that the contact may be sixpointic. The formulae thence arising turn out to be strictly analogous to those belonging to the case of three variables, except that the arbitrary quantities cannot in general be divided out from the final expression. In fact, it is the presence of these quantities which enables us to determine the position of the plane of section, and the equation whereby this is effected proves to be of the degree 10 in : 't'=-k, and besides this of the degree 12n-27 in the coordinates x, y, z, t (n being the degree of the surface), giving rise to the theorem above stated. Beyond the question of the principal tangents, it has been shown by Clebsch and Salmon that on every surface U a curve may be drawn, at every point of which one of the principal tangents will have a fourpointic contact. And if n be the degree of U, that of the surface S intersecting U in the curve in question will be 1 ln--24. Further, it has been shown that at a finite number of points the contact will be five-pointic. The number of these points has not yet been completely determined; but Clebsch has shown (Crelle, vol. lviii. p. 93) that it does not exceed n(lln-24)
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