MESSENGER OF MATHEMATICS. A SMITH'S PRIZE DISSERTATION. By Professor Cayley. WRITE dissertation: On the general equation of virtual velocities. Discuss the principles of Lagrange's proof of it and employ it [the general equation] to demonstrate the Parallelogram of Forces. Imagine a system of particles connected with each other in any manner and subject to any geometrical conditions, for instance, two particles may be such that their distance is invariable, a particle may be restricted to move on a given surface, &c. And let each particle be acted upon by a force [this includes the case of several forces acting on the same particle, since we have only to imagine coincident particles each acted upon by a single force]. Imagine that the system has given to it any indefinitely small displacement consistent with the mutual connexions and geometrical conditions; and suppose that for any particular particle the force acting on it is P, and the displacement in the direction of the force (that is, the actual displacement multiplied into the cosine of the angle included between its direction and that of the force P) is = 8p. Then Sp is called the virtual velocity of the particle, and the principle of virtual velocities asserts that the sum of the products Pop, taken for all the particles of the system, and for any displacement consistent as above, is = 0; say that we have ΣΡδρ = 0. VOL. III. B The Treatment of Proportion by the Association for the Improvement of Geometrical Notes. By H. M. Taylor 155 164 ANALYTICAL GEOMETRY OF TWO AND THREE DIMENSIONS. Problem. By Professor Cayley On a Formula in the Geometry of the Sphere. By R. F. Scott 50 58 On Residuation in Regard to a Cubic Curve. By Professor Cayley 62 The Collection of Models of Ruled Surfaces at South Kensington 111 Geometrical Investigation of Some Properties of Quadric Surfaces. By 122 Cartesian Ovals Regarded as Projected Intersections of Surfaces of the 141 ARITHMETIC, ALGEBRA, AND TRIGONOMETRY. Relations between the Angles of Regular Bodies. By S. M. Drach Logarithmic and Factor Tables. By S. M. Drach 6 Remarks on Logarithmic and Factor Tables, with special reference to 7 12 Notes on Conterminal Angles. By W. D. Bushell On Ludolff Van Ceulen's 35-decimal value of and on some of his works. On the Quadrature of the Circle, A.D. 1580-1630. By J. W. L. Glaisher On Periods in the Reciprocals of Primes (second paper). By W. Shanks Generalisation of Mr. Holmes's Theorem. By J. W. L. Glaisher PAGE On the Probability of errors of different amounts in Calculations, attributable to the uncertainty of the last figure through contraction. By J. W. L. Glaisher On Mr. Ambrose Smith's Experimental Determination of the Value of by the Theory of Probability (vol. II., p. 119) On the Calculation of Logarithms. By H. Wace Professor Hermite's Proof of the Irrationality of ex when x is an Integer Mr. Hanlon's suggestions towards the formation of an extended Table of 100 Note on a Question in Probabilities connected with the performance of Calculations in Duplicate. By J. W. L. Glaisher A Continued Fraction for tan nx. By J. W. L. Glaisher A Formula in Trigonometry. By H. M. Taylor DIFFERENTIAL AND INTEGRAL CALCULUS. 106 137 163 Exercises in the Integral Calculus, No. II. By Sir James Cockle 108 Review of Todhunter's Researches on the Calculus of Variations. By M. M. 184 MECHANICS AND HYDROSTATICS. 120 138 144 Note on Hydrodynamics. By E. J. Nanson On the Impact of Elastic Rods. By E. J. Nanson On the Motion of a Particle toward an Attracting centre at which the force is Infinite. By A. Hall MISCELLANEOUS. A Smith's Prize Dissertation. By Professor Cayley A Smith's Prize Paper and Dissertation; Solutions and Remarks. By TRANSACTIONS OF SOCIETIES, &c. 165 The Meeting of the British Association at Bradford. By J. W. L. Glaisher, MESSENGER OF MATHEMATICS. A SMITH'S PRIZE DISSERTATION. By Professor Cayley. WRITE dissertation: On the general equation of virtual velocities. Discuss the principles of Lagrange's proof of it and employ it [the general equation] to demonstrate the Parallelogram of Forces. Imagine a system of particles connected with each other in any manner and subject to any geometrical conditions, for instance, two particles may be such that their distance is invariable, a particle may be restricted to move on a given surface, &c. And let each particle be acted upon by a force [this includes the case of several forces acting on the same particle, since we have only to imagine coincident particles each acted upon by a single force]. Imagine that the system has given to it any indefinitely small displacement consistent with the mutual connexions and geometrical conditions; and suppose that for any particular particle the force acting on it is P, and the displacement in the direction of the force (that is, the actual displacement multiplied into the cosine of the angle included between its direction and that of the force P) is Sp. Then dp is called the virtual velocity of the particle, and the principle of virtual velocities asserts that the sum of the products Pop, taken for all the particles of the system, and for any displacement consistent as above, is = 0; say that we have = ΣΡδρ = 0. VOL. III. B |