BIBLIOGRAPHICAL NOTES. 513 



&quot;The number w&quot; which he sees does not depend on the choice of origin &quot;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.&quot; The vector 

 part of the quaternion is the vector &quot;perpendicular let fall from the assumed 

 origin on the central axis of the system&quot; (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, 289336 (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. 



SCHO NEMANN (T.) Ueber die Construction von Normalen und Normalebenen ge- 

 wisser krummer Fldchen und Linien. Monatsberichte der koniglichen 

 preussischen Akademie der Wissenchaften fiir 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). Schonemann 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. 3943, 1881). In our language the two rays are the two screws 

 of zero pitch on the cylindroid 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 Mathematiques, 2 e Ser., Vol. xl., 1866. 



WBIBKSTRASS (C.). Ueber ein die homogenen Functionen zweiten grades betreffendes. 

 Theorem nebst Anwendung desselben auf die Theorie der kleinen Schwin- 

 gungen. Monatsberichte der k. preus.sischen Akademie der Wissenschaften, 

 1858, pp. 207-220; and Mathematische Werke, Vol. i. pp. 233-246. 



Let &amp;lt;, ij/ be two homogeneous quadratic functions of n variables x lt ... x n and 

 lety*(s) be the discriminant of s&amp;lt;$&amp;gt; - ^/. 



If the discriminant of one of the functions, say &amp;lt;f&amp;gt;, does not vanish, and if 

 further &amp;lt;J&amp;gt; is essentially one-signed vanishing only when all the variables vanish, it 

 can be shown that s lt s 2 , ... s n the roots of/(s) = (assumed distinct) are all real 

 and &amp;lt;, (// can then be reduced to the forms 



where y^...y n are all real linear functions of x l ... x n and e is + 1 according as 

 positive or negative. See 86 and p. 484. 



B. 33 



