528 THE THEORY OF SCREWS. 



BUCHHBIM (A.) On the Theory of Screw* in Elliptic Space. Proceedings of the 

 London Math. Soc., Vol. xiv., p. 83; Vol. xvi., p. 15; Vol. xvii., p. 240; 

 Vol. xviii., p. 88. 



In these papers the methods of the Ausdehnungslehre of Grassmann have been 

 applied to the Biquaternions of Clifford. Reference should also be made to another 

 paper by the same author, &quot;A Memoir on Biquaternions.&quot; (American Journal of 

 Mathematics, Vol. vii., No. 4, p. 23, 1884.) If A, B be two biquaternions they 

 determine a linear singly infinite series of biquaternions \A + p.B when X, p. are 

 scalars : this set is called a cylindroid, so that if (7 is any biquaternion of the 

 cylindroid -4, B we have C = \A+p.B. A remarkable investigation of the 

 equation to this surface in elliptic space is given, and a generalization of the plane 

 representation of the cylindroid is shown. In these writings it is the methods 

 employed that are chiefly noticeable. We find however much more than is implied 

 by the modest disclaimer of the lamented writer, who in the last letter I had from 

 him says, &quot;I have been but slaying the slain, i.e. discovering over again results 

 obtained by you and Clifford.&quot; 



SBGRE (C.) Sur une expression nouvelle du moment mutuel de deux complexes 

 lineaires. Kronecker s Journal, pp. 169-172 (1885). 



A remarkable form for the expression of the virtual coefficient of two screws on 

 a cylindroid is given in this paper. Translated into the terminology of the present 

 volume we can investigate Segre s theorem as follows. 



Let two screws on the cylindroid make angles 0, &amp;lt;, with one of the principal 

 screws, while the zero pitch screws make angles + a, a. Let p be the anharmonic 

 ratio of the pencil parallel to these four screws so that 



sin (6 a) sin (&amp;lt;j&amp;gt; + a) 

 sin (6 + a) sin (&amp;lt; a) 

 Then as usual 



Pe = Po + m cos 2 #&amp;gt; 

 = p -f m cos 2a ; 

 whence 



p 6 = 2m sin (a 0) sin (a + 6), 



p$ ~ 2m sin (a &amp;lt;/&amp;gt;) sin (a + &amp;lt;) ; 

 whence 



4m 2 sin 2 (6 a) sin 2 (&amp;lt;p + a) = 

 Thus 



2m sin (0 a) sin (c + a) = J p ^ 



2m sin (6 + a) sin (&amp;lt;/&amp;gt; a) = Jp~ 1 

 adding, we easily obtain 



If therefore we make 



P = e 216 , 

 we have as the result Segre s theorem that 



D EMILIO (R). Gli assoidi nella statica e nella cinematica. Nota su la teoria delle 

 dinami. Atti del Reale Istituto Veneto di scienze, (6) iii. 1135-1154 



(1885). 



This is an account of the fundamental laws of the different screw-systems. 



