PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY. VOL. IV. PART II [LENT AND EASTER TERMS, 1881.] CONTENTS: PAGE Mr A. G. GREENHILL, Determination of the greatest height consistent with stability that a vertical pole or mast can be made, and of the greatest height to which a tree of given proportions can grow 65 Mr J. N. LANGLEY, On the estimation of ferment in gland-cells by means of osmic acid 74 Dr W. H. GASKELL, On the Action of the Vagus Nerve upon the frog's heart 75 Mr A. G. GREENHILL, On Conjugate Functions of Cartesians and other 77 Dr J. B. PEARSON, On the probable secular change in the position and aspect of the Constellation Ursa Major 93 Prof. CAYLEY, On the elliptic-function solution of the equation 23+y3-1=0 106 109 110 Sir G. B. AIRY, Continued observations on the state of an eye affected with a peculiar malformation. Dr C. 8. Roy, On the mechanism of the renal secretion Cambridge: PRINTED AT THE UNIVERSITY PRESS. AND SOLD BY DEIGHTON, BELL AND CO. AND MACMILLAN AND CO. CAMBRIDGE ; BELL AND SONS, LONDON. PROCEEDINGS OF THE Cambridge Philosophical Society. February 7, 1881. PROFESSOR NEWTON, PRESIDENT, IN THE CHAIR. Horace Darwin, M.A., Trinity College, was balloted for and duly elected a Fellow of the Society. The following communication was made to the Society: Determination of the greatest height consistent with stability that a vertical pole or mast can be made, and of the greatest height to which a tree of given proportions can grow. By A. G. GREENHILL, M.A. I. Suppose a uniform cylindrical pole or wire fixed in a vertical direction at its lowest point, and carried to such a height that the vertical position becomes unstable and flexure begins; it is required to determine this height. Let 2a be the diameter in inches, and A the sectional area of the pole in square inches: and E be Young's modulus of elasticity of the substance, expressed in gravitation measure of Hb to the square inch. Then, if p be the radius of curvature of the central fibre of the pole, the bending moment of resilience (the unit being the inch-lb.) Take the origin O at the top of the pole in its vertical position and the axis Ox directed vertically downwards: then if APB be the central line of the pole, supposed to be slightly dy 1 dp Ρ dx This must be equated to the moment of the weight of PB about P, which x', y' denoting the co-ordinates of any point Q between B and P, and w the density of the substance in pounds to the cubic inch. Therefore the differential equation of the central line APB is To solve this differential equation, first put paz, then where 2= AJ (kr) + BJ_(Kr), Ꮖ . T J2(x) = √π2*T(n+})Jo √π2′′ (n+b) cos (x cos p) sinTM þdø. (Todhunter, Functions of Laplace, Lamé, and Bessel, p. 285.) Consequently the solution of (2) is At 4, the lowest point, we must have p= 0; and therefore, supposing the height ỔA to be h, then J_4(xh®) = 0. If c be the least root of the equation J_(c) = 0, |