A particular class of dynamical problems has recently been treated geometrically by Sir Robert Stawell Ball, formerly
astronomer royal of Ireland, now Lowndean Professor of Astronomy and Geometry at Cambridge. His method is given in a work entitled Theory of Screws, Dublin, 1876, and in
subsequent articles. Modern geometry is here drawn upon, as was done also by Clifford in the related subject of Biquaternions.
Arthur Buchheim of Manchester (1859–1888),
showed that Grassmann's Ausdehnungslehre supplies all the
necessary materials for a simple calculus of screws in elliptic space. Horace Lamb applied the theory of screws to the question
of the steady motion of any solid in a fluid.
Advances in theoretical mechanics, bearing on the integration and the alteration in form of dynamical equations, were made since Lagrange by Poisson, William Rowan Hamilton,
Jacobi, Madame Kowalevski, and others. Lagrange had
established the "Lagrangian form" of the equations of motion. He had given a theory of the variation of the arbitrary constants which, however, turned out to be less fruitful in results than a theory advanced by Poisson.99 Poisson's
theory of the variation of the arbitrary constants and the method of integration thereby afforded marked the first onward step since Lagrange. Then came the researches of Sir William Rowan Hamilton. His discovery that the integration
of the dynamic differential equations is connected with the integration of a certain partial differential equation of the
first order and second degree, grew out of an attempt to deduce, by the undulatory theory, results in geometrical optics previously
based on the conceptions of the emission theory. The Philosophical Transactions of 1833 and 1834 contain Hamilton's papers, in which appear the first applications to mechanics of the principle of varying action and the characteristic
function, established by him some years previously. The object which Hamilton proposed to himself is indicated by the title of his first paper, viz. the discovery of a function by means of which all integral equations can be actually represented. The new form obtained by him for the equation of motion is a result of no less importance than that which was the professed object of the memoir. Hamilton's method of integration was freed by Jacobi of an unnecessary complication,
and was then applied by him to the determination of a geodetic line on the general ellipsoid. With aid of elliptic co-ordinates