The equations governing the motion of a viscous fluid were first obtained by Navier more than 100 years ago (“ Memoire sur les Lois du Mouvement des Fluides,” Mem. de l’Acad. des Sciences, vi. 389, 1822), and in spite of their close study by Stokes, Helmholtz, Kelvin, Rayleigh, Lamb and numerous other mathematicians of great eminence, no complete unrestricted solution for any case of practical importance has yet been discovered. As they stand, the mathematical difficulties presented by the equations have so far been found to be too formidable, and whatever progress has been achieved has been by imposing restrictions on the form of the equations, and therefore serious limitations on the nature of the fluid motion studied. It may be that the mathematical symbolism in the formulation of the problem of viscous fluid flow as usually presented is not that best adapted for its purpose, that it is not as natural a medium of expression as for example Tensor Analysis is for Relativity ; that in fact the essential factors that govern the eddying, for instance, in the wake of a moving body are not presented as governing the structure of the equations.

Elsewhere I have shown how the study of compound interest leads to the conception of a “growth curve” (the exponential curve) from whose properties we can derive, first, the theory of logarithms, then the conception of e as the limit of (1 + 1/n)n, and finally the differential coefficients of ax and logax. The method of differentiating ax and of introducing e to be described here follows entirely different lines, and may be useful to students who have approached logarithms in the ordinary way through the theory of indices and have begun the study of the calculus before reading what is commonly called “higher algebra.”

The theory of the orthocentre is usually considered in conjunction with ordinary triangles only, that is to say, with triangles whose three vertices or sides are distinct. The treatment for the case when two or more sides or vertices coincide is generally overlooked, but it leads to many interesting results and applications. An exposition of some of these is the object of this note.

If a rod of fixed length a slides with its ends on two fixed lines at right angles, its envelope is an astroid or hypocycloid of four cusps whose equation referred to the two fixed lines as axes is .

If the two lines are not at right angles but inclined at an angle 2a (a < 46°), the envelope of the rod is a curve with four eusps in shape like an astroid elongated along one of the bisectors of the axes. This curve can be shown to be an involute of the astroid.