The problem referred to is that of inscribing in a sphere a closed n-sided polygon, whose sides shall pass respectively through n given points which are not on the surface. Hamilton evidently regarded his solution of this question as a very tough piece of mathematics (see his Life, vol. iii. pp. 88, 426). In preparing a new edition of my Quaternions, I was led to a mode of treating this question which enables us to dispense with the brilliant feats of analysis which seem to be required in Hamilton's method.