Alfred George Greenhill
In the mathematical treatment of Wave Motion we are constrained at present to employ the approximation of supposing the velocities of the liquid particles due to the wave motion to be sufficiently small for the squares, etc., of the particles velocities to be neglected; although it is singular that this approximation is not required in the first problem of wave motion ever solved, discovered by Gerstner in 1802, and afterwards independently by Rankine in 1862 (Stokes, Mathematical and Physical Papers, I, p. 219).
- (1886). "Wave motion in hydrodynamics". American Journal of Mathematics 9: 62–96. (quote from p. 62)
"The Applications of Elliptic Functions" (1892)
Previously to the introduction of the Elliptic Functions, the Circular Pendulum could only be treated by means of the circular functions, by considering the oscillations as indefinitely small, and by assimilating its motion to that of Huygens' Cycloidal Pendulum, of 1673.
Previously to Abel's discovery (1823) it was the elliptic integral which was studied, as in the writings of Euler and Lagendre; and, in fact, in a physical and dynamical problem it is the elliptic integral which arises in the course of the work; for instance in the form of the Equation of Energy,
½(dx/dt)2=X, so that √2 t=∫dx/√X
and now, when X is a cubic or quadratic function of x, so that (dx/dt)2 is a quadratic or cubic ... the integral is called an elliptic integral of the first kind; and we have to follow Abel and determine the elliptic function which expresses x as a function of t.
The Theory of Transformation may be developed entirely from the algebraical point of view; but Abel has shown how the form of the transformation of the nth order may be inferred from the elliptic functions of the nth parts of the periods, called by Klein, modular functions.
Elliptic functions are doubly periodic. A function of a single variable cannot have more than two distinct periods, one real and one imaginary, or both complex. For if a third period was possible, the three sets of period parallelograms obtained by taking the periods in pairs would reach every point of the plane, so that the function would have the same value all points of the plane, and would therefore reduce to a constant (Bertrand, Calcul intégral, p. 602).
- Chapter 10, p. 338 (remark translated from H. A. Schwarz's Formeln und Lehrsätze zum Gebrauche der elliptischen Funktionen)
"A Treatise on Hydrostatics" (1894)
The Science of Hydrostatics is considered to originate with Archimedes (B.C. 250) in his work Περὶ ὀχουμένων now lost, but preserved in the Latin version of Guillaume de Moerbek (1269), "De iis quœ vehuntur in humido"; and recently translated into French by Adrien Legrand, "Le traiteé des corps flottants d'Archimède," 1891.
Consider now the pressure in liquid which is moving bodily in a vessel with given acceleration a in a fixed direction, like the water in the boiler or tender of a locomotive engine. It is convenient to reduce Dynamical problems to a question of Statics by the application of D'Alembert's principle, which asserts that "the reversed effective and impressed forces of a system are in equilibrium," the effective force of any particle or body being defined as the force required to give it the acceleration which it actually takes.
"Report on Gyroscopic Theory" (1914)
These principles require to be translated into the equations of a dynamical problem, expressed either in the abridged vectorial system in accordance with the modern procedure of Maxwell, Clifford, Gibbs, Burali-Forti and Marcolongo, Föppl, Jahnke, Gans, Silberstein, and others; or as formerly, with the ordinary scaffolding of Cartesian and Eulerian co-ordinates.
The article by Stäckel, Elementary Dynamics, in the Math. Encyclopædia will give the historical notices of the writers quoted: Newton, Euler, Segner, Dalembert, Lagrange, Poinsot, Hamilton, Maxwell, Clifford.