ON GRAPHIC METHODS IN MECHANICAL SCIENCE. 403 



instrument works with great smoothness, as, with the exception of the 

 driving bevel wheels, it consists entirely of linkvvork. At the end of the 

 paper mechanism is suggested for continuously describing bi-circular 

 quartics. 



The actual problems solved by means of the ellipse will be alluded to 

 when dealing with the ellipse of stress and other graphical methods. 



The parabola is another important curve in graphical methods, the 

 power of drawing which is indispensable in every problem of bending 

 moment on girders subjected to any uniform or travelling load, and for 

 many other problems. From the fact that all parabolas are similar, it is 

 a simple matter to have parabolic curves of cardboard or other material 

 of various sizes, and, having used whatever size may be most convenient, 

 to find the constants for the particular problem. The authors above- 

 mentioned have designed a mechanism, points in which simultaneously 

 describe parabolas required in problems of bending moment.' This 

 mechanism consists of a board, slot frame, compass bars, transverse bar, 

 distance-piece, and distorting-table. As the slot frame moves across the 

 face of the board all points in the mechanism describe the same parabola 

 in different positions. By means of this model bending moment can be 

 clearly demonstrated for any point of the span, whether for a concentrated 

 load or for a load distributed over several points, as in the case of a rail- 

 way train, either longer or shorter than the span of the bridge or girder 

 under consideration,^ 



Professor H. J. S. Smith mentions the apparatus of Mr. A. E. 

 Donkin for tracing harmonic curves — a machine drawing as many 

 difEerent forms of the curve y=a sin (')nx + A) + b sm(nx + B') as there 

 are means for varying the constants a, b, m, n, A, B, the number of 

 variations being practically unlimited, except in the case of m and n, 

 which are the number of teeth in certain wheels, which therefore admit of 

 only a limited number of changes. 



The most important carves used in the teeth of wheels are the 

 involute, cycloid, epicycloid, and hypocycloid. Practical modes of de- 

 scribing these are given by Professor Unwin.^ 



Another curve which has important properties, and which may be used 

 for the extraction of roots, is the logarithmic spiral ; it ma^y also be sub- 

 stituted for table logarithms, as mentioned by Professor Favaro in his 

 book on graphic statics. The same author describes the gaharit of 

 Professor Steiner,'* which is an instrument for facilitating the construction 

 of logarithmic spiral, and may be applied to graphic calculations. The 

 centre of the curve can be marked upon the paper by means of a pricker 

 which passes thi'ough a small hole. The apparatus has, according to 

 Professor Favaro, the following uses : — 



1. The curve presents a series of curvatures of which the radii cross 



' Moment Delineator, by Professor Alexander, M.A., and Mr. Arthur Thomson, 

 B.Sc. Institution of Engineers and Shipbuilders in Scotland. Read November 19, 

 1S89. 



- A clear elementary treatment of the use of the parabola in problems relating to 

 girders is given in the Ajijjlied MeohaiUcs of Messrs. Alexander and Thomson. 

 Macmillan it Co. 



* Elements of Machine Denign, by Professor Unwiu, B.Sc. Longmans, Green, 

 & Co. 



" Leqons de StMiqne grapliiqve, par Antonio Favaro. Traduites do I'italien par Paul 

 "Terrier. Gauthicr-Villars, Paris, 1885. 



D n 2 



