A Treatise on the Theory of Screws

BIBLIOGRAPHICAL NOTES. 513 “The number w” which he sees does not depend on the choice of origin “will denote the (real) quotient obtained by dividing the moment of the principal resultant couple by the intensity of the resultant force', with the known direction of which force the axis of this principal (and known) couple coincides, being the line which is known by the name of the central axis of the system.” The vector part of the quaternion is the vector “ perpendiculai- let fall from the assumed origin on the central axis of the system” (p. 40). It is interesting to note that the scalar w is what we would term the pitch of the Screw on which the wrench acts. Poinsot (L.)—Theorie nouvelle de la rotation des corps. Liouville’s Journal Math.; Vol. xvi., pp. 9-129, 289—336 (March, 1851). This is Poinsot’s classical memoir, which contains his beautiful geometrical theory of the rotation of a rigid body about a fixed point. In a less developed form the Theory had been previously published in Paris in 1834, as already mentioned. Schönemann (T.)—Ueber die Construction von Normalen und Normalebenen ge- wisser krummer Flächen und Linien. Monatsberichte der königlichen preussischen Akademie der Wissenchaften für das Jahr 1855, pp. 255-260. Believing that this paper was but little known Herr Geiser reprinted it in Crelle’s Journal, Vol. xc., pp. 44—48 (1881). Schönemann there gave the im- portant theorem which has since been independently discovered by others, namely that whenever a rigid body is so displaced that four of its points, A, B, C, D move on fixed surfaces the normals to the surfaces which are the trajectories of all its points intersect two fixed rays. Herr Geiser gives an analytical proof (Crelle, Vol. xc., pp. 39—43, 1881). In our language the two rays are the two screws of zero pitch on the cyliudroid reciprocal to the freedom of the body, and the cylindroid is itself determined by being reciprocal to four screws of zero pitch on the normals at A, B, C, D respectively to the four fixed surfaces. Another proof is given by Ribaucour, Comptes rendus, Vol. Ixxvi., p. 1347 (2 June, 1873). See also Mannheim (A.), Liouville’s Journal de Mathématiques, 2e Sér., Vol. xl., 1866. Weierstrass (C.). Ueber ein die homogenen Functionen zweiten grades betreffendes. Theorem nebst Anwendung desselben auf die Theorie der kleinen Schwin- gungen. Monatsberichte der k. preussischen Akademie der Wissenschaften, 1858, pp. 207-220; and Mathematische Werke, Vol. i. pp. 233-246. Let </>, f be two homogeneous quadratic functions of n variables xlt ... xn and let f (s) be the discriminant of s</> - i//. If the discriminant of one of the functions, say </>, does not vanish, and if further </> is essentially one-signed vanishing only when all the variables vanish, it can be shown that s2, ... sn the roots of f (s) = 0 (assumed distinct) are all real and <f>, i// can then be reduced to the forms 4> = e («/?+ +2//) <A = «(«i2/i3+ ••• s»3/n2) where yx ... yn are all real linear functions of ajx ... xn and e is + 1 according as </> is positive or negative. See § 86 and p. 484. b. 33 ■