SECTION MECHANIC 9. 



-i any other properties of these curves here ; many gentlemen here 

 must be quite familiar with the proofs. I must content myself with 

 simply showing you, by way of illustration, models of mechanisms in 

 which the centroids are constructed in such a way that their rolling can 

 be distinctly seen. By the aid of these centroids we can treat a great 

 number of kinematic and dynamic problems in an extremely simple 

 and beautiful manner, and can treat complicated and simple prob- 

 lems by one general method, instead of using different methods. 

 Centroids have not, so far as I know, been used in English text 

 books hitherto, but have not unfrequently found more or less 

 extended use in German and French works, principally in static 

 and kinematic problems. I have found them, however, even more 

 useful in Kinetics. Among the more recent books in which I have 

 noticed them, I may mention Dwelshauver's Dery's "Cine'matique," 

 Schell's "TheoriederBewegungundder Krafte,"and Prdll's"Graphische 

 Dynamik." 



Considering the constrained motion of bodies instead of plane figures 

 only, the instantaneous centre becomes an axis, and the centroidal 

 curve a ruled surface, the locus of all the axes. These surfaces are 

 called axoids. For bodies having complane motion, they are 

 cylinders ; for bodies having motion about a fixed point, cones ; and 

 for general motion in space, general ruled surfaces, in general non- 

 developable, of which successive generators twist upon each other. 

 Poinsot was, I believe, the first to mention these surfaces.^ 



I shall only mention one more point in connection with the subject 

 before us. Professor Reuleaux has devised, for the purpose of aiding 

 his kinematic work, a system of notation founded upon his analysis, 

 which can be used to represent these mechanisms in a perfectly simple 

 manner. Of course, to write down a description of them is a long 

 matter, but hitherto we have not known their real nature analytically, 

 and therefore could do nothing else. Now that we do know it, we 

 may treat them exactly as we do chemical compounds, where instead 

 of writing down the whole names of everything, we use short symbols 

 for known elements. This kinematic notation I can do no more than 

 mention. It seems to me very original, and I have myself found it 

 very useful in the analysis of really complicated machinery. 

 Mechanisms of great apparent complexity come out, often, in forms of 



