A problem of frequent occurrence in the analysis of engineering structures is the determination of the resultant actions in a uniform bar which is subjected simultaneously to an axial thrust and to lateral loads and terminal couples. This type of problem can be solved by straightforward analytical methods, and well-known formulas for certain cases of loading have been obtained by this means. An alternative method of solution is, however, available by the use of a polar diagram originally due to Ratzersdorfer but discovered independently by H. B. Howard and introduced by him to British engineers. This method is of great, value in obtaining graphical solutions in numerical cases and has been used very widely in drawing offices for this purpose. In addition to its use for the solution of specific problems, however, it offers an attractive method of deducing general formulas and is likely to appeal to the imagination of many students far more than the usual approach. This application of the diagram does not appear to have received the attention that it deserves and the object of this paper is to show its value to the teacher.

Writers on algebra have noticed that the usual laws of indices lead to self contradictions (Chrystal called them “paradoxes”), no matter whether the meaning of a fractional index is defined by

or by

but it does not appear that anyone has yet shown how the laws can be modified in a way that makes them self-consistent. In this note, the necessary amendments are stated for each of these definitions, and are shown to be very much simpler for definition (1). This is fortunate; for that definition appears to be the generally accepted one, though definition (2) is adopted by Hardy and some others.