municated to the British Association a method of drawing the reciprocal figure, founded on the theory of reciprocal polars. I have since found that the construction of diagrams of forces, in which each force is represented by one line, had been independently discovered by Mr. W. Taylor and had been used by him as a practical method of determining the forces acting on frames for several years before I had taught it in King's College or even studied it" (Maxwell's Scientific Papers, vol. ii. pp. 163, 164, edited by W. Niven, M.A., F.R.S.) The simple demonstration given in § 49 of this work, where the reciprocity of the force and funicular polygons is derived without the aid of Maxwell's theorem or the Nullsystem, is here published as a whole for the first time. The work, although of comparatively small dimensions, is the fruit of much labour, and represents an honest effort to popularise a most useful branch of geometry, by illustrating its varied application to such practical problems as finding the most economical section for a rail or girder, and the relative angle of advance of the expansion over the main excentric arm in valve-gear design. R. H. G. CONTENTS CHAPTER I.-ANHARMONIC PENCILS AND RATIOS § 1. Definition of Projectivity 2. Harmonic Properties of Quadrangle |