TRANSACTIONS OF SECTION A. 559 



remembered that the action of the instrument is, as it were, arithmetical rather 

 than geometrical, for a cam is useful only with reference to its own scale. 



This instrument not only enables transformations of a definite and known 

 character to be made, but is equally applicable for transforming in an empirical 

 manner. The curve drawn by a recording -voltmeter or ammeter may thus be 

 replotted for estimation of area, or other graphical analysis, without any knowledge 

 of the law of the particular instrument. In other words, a correction can be 

 applied to a curve. 



The cams are easy to make, and even if carelessly cut cannot possibly give 

 rise to cumulative errors. It is convenient to use the upper edge of the ruler 

 instead of the edge which rolls on the cam. The curve must in this case be set 

 out with the ruler, and used with the same ruler, or one of the same width. The 

 rolling of a straight edge on a cam has been used in a photometer, invented by 

 Mr. W. H. Preece and the author,^ for the automatic calculation of the squares of 

 the displacements of a lamp. 



3. On a Linkage for the Automatic Description of Regular Polygons. 

 By Professor J. D. Everett, F.R.S. 



Let any number of equal bars be jointed together in the manner of a lazytongs, 

 so as to lie in two superposed planes, each bar in one plane (except the end bars) 

 being jointed at both ends, B, D, and at one intermediate point, C, to the correspond- 

 ing points of three bars in the other plane ; but instead of the two distances BC, CD 

 being equal, as in the ordinary lazytongs, let them be unequal, CD being the greater. 

 All the bars are to be precisely alike. They will form a frame with one degree of 

 freedom, resembling in this respect an ordinary lazytongs ; but instead of the three 

 series of points, Bj Bj . . . , Cj C, . . ., Dj D^ . . ., being ranged in three parallel 

 straight lines, they will be ranged in three concentric circular arcs, two of which, 

 namely, B, Bj . . . and Dj D., . . . , formed by the iuner and outer ends respec- 

 tively, will subtend the same angle at the common centre O. In place of thp 

 rhombuses of the ordinary lazytongs we shall have kites, and the axes of all the 

 kites will pass through O. 



When the frame, supposed to be at first pushed close in, is gradually opened out 

 so as to increase the widths and diminish the lengths of the kites, the curvatures will 

 increase in a double sense : the arcs formed by the inner and outer ends will in- 

 crease in length, and at the same time their radii will diminish. The common 

 angle which they subtend at the centre will accordingly increase very rapidly, and 

 may easily amount to 360° or more. When it is exactly 360°, the first and last 

 points B will coincide, as will also the first and last points D. In this position the 

 points D will be the corners of one regular polygon, the points C of another, and the 

 points B of a third. We have thus an automatic arrangement for constructing a 

 regular polygon with any number of sides. Also, as the axes of successive kites are 

 equally inclined to one another, we have the means of dividing an arbitrary angle 

 into any number of equal augle.s — an end which can also be attained by employing 

 the principle that equal arcs in a circle subtend equal angles at a point on the cir- 

 cumference, or, still more conveniently, by making use of the fact thar, those bar.s 

 which correspond to parallel bars in an ordinary lazytongs are equally inclined each 

 to the next. 



Strictly speaking, the figures obtained are not polygons, but stars, which can be 

 converted into regular polygons by joining the ends of their rays. It frequently 

 happens that the curvature can be extended far beyond .S60°, giving a succession of 

 regular stars with a continually decreasing number of rays. 



Let each of the bars above described be lengthened at its inner end, B, till a 

 point A is reached, such that the three distances, AB, AC, AD, are in geometrical 

 progression. Then it can be shown that the radius OA of the circular arc formed 

 by the ends A is constant, and equal to AC. Hence the common centre, 0, can be 

 found automatically by employing two additional bars of length AC, jointed 



' Proc. Inst. C.E., vol. ex. p. 81. 



