NOTEF. 591
hundred times that the students, in spite of the reiterations of their teachers that no logical difference between the things exist, do make such a dif- ference. I hope it may still not be too late to return to a correct nomenclature. If at any time it be necessary to distinguish between the two curves, it will be both correct and sufficient to do so by calling one the stationary and the other the moving centroid.
10 (P. 75.) The base circles used in drawing the profiles of involute teeth are secondary centroids of this kind. The third centroid is the straight line, rolling on these centroids, of which a point generates the profiles.
11 (P. 80.) [The theorem is contained in Poinsot's celebrated memoir Theorie nouvelle de la Rotation des Corps, presented to the French Institute in 1834, and afterwards published in an extended form in Liouville's Journal, vol. xvi. pp. 9-129 and 289-336 (March, 1851). A translation (by Whitley) of the original paper was published at Cambridge in 1834. The theorem referred to is stated as follows : " The most general motion of which a body is capable is .... that of a certain external screw which turns in the corresponding internal screw." The same paper contains also the first enunciation and proof of the theorem : " the rotatory motion of a body about an axis which incessantly varies its position round a fixed point is identical with the motion of a certain cone whose vertex coincides with this point, and which rolls, without sliding, on the surface of a fixed cone having the same vertex." The simple treatment of such a probleiu which we now adopt was not possible forty years ago ; Poinsot's reasoning included the ideas of force and velocity, instead of merely the notion of change of position.]
12 (P. 80.) \Traitede Cinematique, 1864. Belanger here re-states Poinsot's theorems of motions in a plane and about a point (pp. 55 and 58), and gives the following theorems respecting general motion in space (p. 59) : " Tout mouvement continu d'un systeme invariable equivaut au roulement d'un c6ne lie au systeme, sur un autre c6ne qui aurait un mouvement de translation dans 1'espace, le premier c6ne etant le lieu geome'trique dans le corps en mouvement, des axes instantanes de sa rotation autour du point choisi ; le second cone etant le lieu des memes axes instantanes dans le systeme de comparaison en translation " .... (p. 79) " II f aut ajouter ici que ce me^me mouvement Equivaut a celui d'une surface re'glee qui, tant li^e au systeme, touch erait continuellement, suivant une gene'ratrice, une autre surface re'gle'e fixe dans 1'espace, sur laquelle elle roulerait en glissant a chaque instant le long de la ge'neratrice de contact des deux surfaces. En effet, les positions successives dans 1'espace, de 1'axe instantane* de rotation et de glissement, forment une surface regime, c'est le surface immobile ; et les positions de ce meme axe instantane, relativement an systeme ou corps mobile, forrnent une autre surface reglee mobile, emportant le corps avec elle."
Prof. Ball points out (Theory of Screws, p. xix., etc.) that the discovery of the theorem given in 12, the " canonical form " of the displacement of a rigid body, is due to Chasles. In view of Ball's most interesting investi- gations, as well as of the more elementary treatment of the subject now adopted in other books, I regret that Prof. Reuleaux adheres to the view taken on page 83, where twisting is expressed as a rolling about two axes, one at an infinite distance. This conception seems rather to add difficulty to, than
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