(1) MANIFOLD adaptations of dynamical reasoning have given rise to specialised treatises of undoubted excellence. Prof. Slate sets himself the task of surveying the common foundation and the common stock of resources of these adaptations, as well as the trend of modern development in dynamics. Six quantities enter into the formulation of fundamental dynamical principles: force, power, and force-moment on one hand, momentum, kinetic energy and moment of momentum on the othen Each triad can be, and has been, used in the enunciations of dynamics. But the enunciations involve the use of reference-frames, leading to the question of the relativity of such frames and the transformation from one frame to another—both when the transformation is that of a mere translation, and when it partakes of the more general, form of a shift and a rotation. The author con-siders these transformations and the chief kinds of co-ordinate systems. Euler's and Lagrange's equations, and their use in the study of the dynamics of a rigid body, conclude a presentation possessing considerable interest and originality. A number of notes are added containing references and further elucidations. (1) The Fundamental Equations of Dynamics and its Main Co-ordinate Systems Vectorially Treated and Illustrated from Rigid Dynamics. By Frederick Slate. Semi-centennial Publications of the University of California.) Pp. ix + 233. (Berkeley: University of California Press, 1918.) (2) Projective Vector Algebra: An Algebra of Vectors Independent of the Axioms of Congruence and of Parallels. By Dr. L. Silberstein. Pp. vii + 78.(London: G. Bell and Sons, Ltd., 1919.) Price 7s. 6d. net. (3) Elements of Graphic Dynamics: An Elementary Text-book for Students of Mechanics and Engineering. By Ewart S. Andrews. Pp. viii + 192.(London: Chapman and Hall, Ltd., 1919.) Price 10s. 6d. net. (4) Differential Calculus for Colleges and Secondary-Schools. By Dr. Charles Davison.(Cambridge Mathematical Series.) Pp. viii + 309. (London: G. Bell and Sons, Ltd.; 1919.) Price 6s. (5) The Analytical Geometry of the Straight Line and the Circle. By John Milne. (Bell's Mathematical Series.) Pp. xii + 243. (London: G. Bell and Sons, Ltd., 1919.) Price 5s.