A Treatise on the Theory of Screws

BIBLIOGRAPHICAL NOTES. 527 Cox (Homersham)—On the application of Quaternions and Grassmann’s Aus- dehnungslehre to different kinds of Uniform Space. Cambridge Philosophical Transactions, Vol. xiii., Part II., pp. 69—143 (1882). So far as the Theory of Screws is concerned the chief result in this paper is the demonstration that the homologue of the cylindroid in non-Euclidian space which Lindemann had already shown to be of the fourth degree may be represented by the equation (^« - pf) (w2 + «2) scy = (1 - PaPß) (æ2 + ff) wz. The function known as the sexiant (§ 230) is here generalized into the corre- sponding function of six screws in non-Euclidian space. It is of course a fundamental theorem that a ray crossing two screws of equal pitch meets the cylindroid again in a third screw which it cuts perpendicularly (§ 22). This is here generalized into the theorem that a transversal across two screws of equal pitch on the cylindroid in elliptic space intersects that surface also in two other generators which are conjugate polars with respect to the absolute. I may take this opportunity to observe that the function —— which enters into the above equation of the surface has an instructive property. If p«. and pp be transformed into P,l + m an(j Ppffff reSpectively, where m is different from 1 + mplk 1 + mpß unity, then the above function is unaltered. Hence it follows that if the pitch p of every screw on a screw-system of the nth order in non-Euclidian space receive the transformation into ------then the screws so altered will still constitute an 1 + mp n-system. Thus we generalize that well-known feature of an «.-system of screws in ordinary space which asserts that if the pitches of the screws in an n-system be augmented by a constant the screws so altered will remain an n-system. (See Proceedings of the Royal Irish Academy, 2nd Series, Vol. iv., p. 256 (1884).) Padeletti (Dino)—Sulla piu semplice forma dell’ equazioni di equilibria di im sistema rigido vincolato. Rendiconto della R. Accademia Scienze Fis. e Mat. di Napoli, Fascicolo 1°, 1883. In this short paper the author discusses separately two different cases of freedom and by the aid of the reciprocal screw-system gives in each case the equations of equilibrium. Heath (R. S.)—On the Dynamics of a Rigid Body in Elliptic Space. Phil. Trans. Part ii., 1884, pp. 281-324. “ The special features of the method employed are the extensive use of the symmetrical and homogeneous system of coordinates given by a quadrantal tetra- hedron, and the use of Professor Cayley’s co-ordinates in preference to the ‘Rotors’ of Professor Clifford to represent the position of a line in space.” The Theory of Screws is considered and the nature of the cylindroid in Elliptic Space discussed. The general equations of motion referred to any moving axes are then found, and in a particular case they reduce to a form corresponding to Euler’s equations. When there are no acting forces these equations are solved in terms of the theta- functions. This paper lias been already cited in §§ 412, 420.