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 An important addition to the theory of the motion of a solid body about a fixed point was made by Madame Sophie de Kowalevski[96] (1853–1891), who discovered a new case in which the differential equations of motion can be integrated. By the use of theta-functions of two independent variables she furnished a remarkable example of how the modern theory of functions may become useful in mechanical problems. She was a native of Moscow, studied under Weierstrass, obtained the doctor's degree at Göttingen, and from 1884 until her death was professor of higher mathematics at the University of Stockholm. The research above mentioned received the Bordin prize of the French Academy in 1888, which was doubled on account of the exceptional merit of the paper.

There are in vogue three forms for the expression of the kinetic energy of a dynamical system: the Lagrangian, the Hamiltonian, and a modified form of Lagrange's equations in which certain velocities are omitted. The kinetic energy is expressed in the first form as a homogeneous quadratic function of the velocities, which are the time-variations of the co-ordinates of the system; in the second form, as a homogeneous quadratic function of the momenta of the system; the third form, elaborated recently by Edward John Routh of Cambridge, in connection with his theory of "ignoration of co-ordinates," and by A. B. Basset, is of importance in hydro-dynamical problems relating to the motion of perforated solids in a liquid, and in other branches of physics.

In recent time great practical importance has come to be attached to the principle of mechanical similitude. By it one can determine from the performance of a model the action of the machine constructed on a larger scale. The principle was first enunciated by Newton (Principia, Bk. II., Sec. VIII., Prop. 32), and was derived by Bertrand from the principle of virtual velocities. A corollary to it, applied in ship-