BIBLIOGRAPHICAL NOTES. 521 



cylindroid, or rather rediscoverers, for neither of them was the earliest discoverer, 

 for as shown in p. 510 the cylindroid was first introduced into science by Sir 

 W. R. Hamilton so long ago as 1830. It is worthy of note that three investi 

 gators, and if I may add my own name a fourth also, following different lines 

 of research, have each been independently led to perceive the importance of this 

 particular surface in various theories of systems of lines. 



In the paper now before us I had developed the doctrine of reciprocal screws 

 which is of such fundamental importance in the theory. I had arrived at this 

 doctrine independently, and not until after the paper was printed did I learn that 

 the essential conception of Reciprocal Screws had been announced by Professor 

 Klein a few months before my paper was read (pp. 17, 18, 520). 



These facts have to be mentioned in explanation of the circumstance that this 

 first paper contains no references to the names of either Pliicker and Battaglini or 

 Hamilton and Klein. 



SOMOFF (J.) Sur les vitesses virtuelles d une figure invariable, assujetties a des 

 equations de conditions quelconques de forme lineaire. St Petersb. Acad. Sci. 

 Bull., xviii., 1873, col. 162184. 



This paper is an important one in the history of the subject. Its scope may 

 be realized from the paragraph here quoted. 



&quot; Dans le memoire que j ai 1 honneur de presenter a 1 Academie je donne un 

 moyen analytique pour determiner les vitesses virtuelles d une figure invariable, en 

 supposant que ces vitesses doivent satisfaire & des equations de condition de la 

 forme geiierale que je vients de citer. Je prends en meme temps en consideration 

 les proprietes des complexes lineaires de Pliicker, auxquel les vitesses virtuelles 

 d une figure invariable sont intimement liees.&quot; 



The analytical development of the Theory of the Constraints which follows is 

 founded upon the conventions proposed by M. Resal in his &quot; Traite de Cinematique 

 pure.&quot; 



M. Somoff studies conditions of constraint which he has generalized from M. 

 Mannheim s &quot; $tude sur le deplacement d une figure de forme invariable&quot; (p. 519). 



It is instructive to read M. Somoff s paper in the light of the Theory of Screws. 

 For example on p. 179 he gives the theorem that every system of &quot; virtual 

 velocities&quot; which satisfies three linear equations can be produced by two rotations 

 around two rays common to the three corresponding linear complexes. In our 

 language we express this by saying that any displacement of a body with three 

 degrees of freedom can be produced by rotation around two screws of zero pitch 

 belonging to the system. This is easily seen, for let be the screw about which 

 the required displacement is a twist. Let &amp;lt; be any other screw of the three- 

 system, then the two screws of zero pitch on the cylindroid (9, &amp;lt;) are two axes of 

 rotation that fulfil the required condition. 



The cases of four and five degrees of freedom are also briefly discussed by 

 Somoff, but without the conception of screw motion which he does not employ the 

 results are somewhat complicated. 



Reference may also be made to Somoff, &quot; Theoretische Mechanik&quot; translated 

 from the Russian by A. Ziwet, Leipzig, 18789. 



CLIFFORD (W. K.) Preliminary Sketch of Biquaternions. Proceedings of the 

 London Mathematical Society, Nos. 64, 65, Vol. iv., pp. 381 395 (12th 

 June, 1873). 



This is one of the modern developments of that remarkable branch of mathe 

 matics with which the names of Lobachevsky and Bolyai are specially associated. 

 A Biquaternion is defined to be the ratio of two twists or two wrenches or 



