[Book Reviews]

Abstract

THE four lectures constituting this little book are worthy of the great occasion which called forth their delivery. Prof. Klein uses the particular dynamical problem of the top as an illustration of the advantages that may be gained by utilising the modern theory of functions in applied mathematics. Instead of being content with analytical processes, he strives to the utmost to give a geometrical form to his formulas, and to make the solution intuitive. He passes beyond the parameters of Euler and Rodrigues to apply to dynamics a system of coordinates which Riemann introduced forty years ago in the discussion of certain geometrical problems. Using also Riemann's method of conformal representation, he gives an insight into the inner nature of elliptic functions, and shows that his new parameters are what he calls “multiplicative elliptic functions”—they miss being doubly periodic by being affected by an exponential factor when t (the time) is increased by a period. By means of these parameters the author attains to a clearer, neater and more complete solution of the problem of the motion of a body about a fixed point than had hitherto been reached, and justly claims that he has resolved the problem into its simplest elements. He also deals with Jacobi's famous theorem, that the motion of the top may be represented by the relative motion of two Poinsot motions (or rotations of a body about its centre of gravity which is fixed).

The Mathematical Theory of the Top. Lectures delivered on the occasion of the Sesquicentennial Celebration of Princeton University.

By Felix Klein. Pp. 74. (New York: C. Scribner's Sons, 1897.)