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• W4250862015 abstract "LONDONMathematical Society, May S.-Lord Rayleigh, F.R.S., president, in the chair.-Mr. Tucker communicated a short account of a paper by Dr. Hirst on the correlation of two planes. In a former paper on the subject (Proceedings, vol. v., p. 40), the nature and properties were described first, of an ordinary correlation satisfying any eight given conditions; secondly, of an exceptional correlation of the first order, possessing either a singular point or a singular line in each plane, and satisfying seven conditions; and thirdly, of an exceptional correlation of the second order, having in each plane not only a singular point but also a singular line passing through that point, and satisfying six conditions. Moreover, the two following numerical relations were established between the (tt,?) exceptional correlations of the first order, with singular points and singular lines respectively, which satisfy any seven conditions, and the (, ?) ordinary correlations, which, besides satisfying these same conditions, possess a given pair of conjugate points or con j ugate lines respectively (2v- + p, 2 =? + ?). It was by means of these,relations that the number of ordinary correlations was determined which satisfy any eight elementary conditions. Before they could be applied, however, the exceptional correlations of the first order which satisfy any seven elementary conditions) had to be directly determined, and this determination not unfrequently necessitated the consideration of the projective pro perties of curves of high order. In the present paper the writer shows that the object just referred to can be attained in a very much simpler manner, by means of two general relations, hitherto unobserved, connecting the number of exceptional correlations of the second order, which satisfy any six conditions, with the numbers of exceptional correlations of the first order which, besides satisfying the six conditions in question, possess a given pair eitner of conjugate points or conjugate lines.-The secretary then read part of a paper by Prof. H. Lamb, of the University of Adelaide, on the free motion of a solid through an infinite mass of liquid. Suppose that we have a solid body of any form immersed in an infinite mass of perfect liquid, that motion is produced in this system from rest by the action of any set of impulsive forces applied to the solid, and that the system is then left to itself. The equations of motion of a body under these circumstances have been investigated independently by Thomson and by Kirchhoff, and completely integrated for certain special forms of the body. The object of the present communication is, in the first place, to examine the various kinds of permanent or steady motion of which the body is capable, without making any restrictions as to its form or constitution; and, in the second, to show that when the initiating impulses reduce to a couple only, the complete determination of the motion can be made to depend upon equations identical in form with Euler's well-known equations of motion of a perfectly free rigid body about its centre of inertia, although the interpretation of the solution is naturally more complex. Free use is made throughout the paper of the ideas and the nomenclature of the theory of screws as developed and established by Dr. Ball.- Herr Weichold (Head-master of the Johanneum, Zittau, Saxony) sent a paper (read in part by the secretary) containing a solution of the irreducible case, i.e., of the problem to express the three roots of a complete equation of the third degree, in the case of all these roots being real, directly in terms of its coefficients, by means of purely algebraical and really performable operations, whose number shall always be limited, except in'the case where all these roots are incommensurable.-Mr. H. Hart made three communications: First On the Kinematic Paradox.-Prof. Sylvester has described a system of Peaucellier's cells, the poles of which all move in a straight line, but two of which not directly connected always remained at a constant distance. Such a result is very easily obtained by means of the following relations connecting six points A, B, C, D, E, F, lying on a straight line. If" @default.
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