A Treatise on the Theory of Screws

BIBLIOGRAPHICAL NOTES. 517 When the forces are fewer than seven, the formulæ admit of a special trans- formation, which expresses certain further conditions which must be fulfilled. This very elegant result may receive an extended interpretation. If P„, Plt P.2, &c., denote the intensities of wrenches on the screws 0, 1, 2, &c. ; and if (12) denote the virtual coefficient of 1 and 2, then, when the formulæ of Mr Spottis- woode are satisfied, the n wrenches equilibrate, provided that the screws belong to a screw complex of the (n— l)th order and first degree. PlÜckek (J.)—Neue Geometrie des Raumes gegründet auf die Betrachtung der geraden Linie als Raumelement. Leipzig (B. G. Teubner, 1868—69), pp. 1-374. This work is of course the principal authority on the theory of the linear complex. The subject here treated is essentially geometrical rather than dynami- cal, but there are a few remarks which are specially significant in our present subject; thus the author, on p. 24, introduces the word “ Dyname Durch den Ausdruck ‘Dyname,’ habe ich die Ursache einer beliebigen Bewegung eines starren Systems, oder, da sich die Natur dieser Ursache, wie die Natur einer Kraft überhaupt, unserem Erkennungsvermögen entzieht, die Bewegung selbst, statt der Ursache die Wirkung, bezeichnet.” Although it is not very easy to see the precise meaning of this passage, yet it appears that a ‘ Dyname ’ may be either a twist or a wrench (to use the language of the Theory of Screws). On p. 25 we read:—“Dann entschwindet das specifisch Mechanische, und, um mich auf eine kurze Andeutung zu beschränken : es treten geometrische Gebilde auf, welche zu Dynamen in derselben Beziehung stehen, wie gerade Linien zu Kräften und Rotationen.” There can be little doubt that the “geometrische Gebilde,” to which Plücker refers, are what we have called screws. As we have already stated (§ 13), we find in this book the discussion of the surface which we call the cylindroid, to which, as pointed out on p. 510, Sir W. R. Hamilton had been previously conducted. Through any point a cone of the second degree can be drawn, the generators of which are lines belonging to a linear complex of the second degree. If the point be limited to a certain surface the cone breaks up into two planes. This surface is of the fourth class and fourth degree, and is known as Kummer’s surface. See papers by Kummer in the Monatsberichte of the Berlin Academy, 1864, pp. 246— 260, and 495-499. It has since been extensively studied from various points of view by many mathematicians. This theory is of interest for our purpose, because the locus of screws reciprocal to a cylindroid is a very special linear complex of the second degree, of which the cylindroid itself is the surface of singularities. Kummer’s surface has in this case broken up into a plane and a cylindroid. Klein (F.)__Zur Theorie der Linien-Complexe des ersten und zweiten Grades. Math. Ann.; Vol. ii., pp. 198-226 (14th June, 1869). The “ simultaneous invariant ” of two linear complexes is discussed. In our language this function is the virtual coefficient of the two screws reciprocal to the complexes. The six fundamental complexes are considered at length, and many remarkable geometrical properties proved. It is a matter of no little interest that these purely geometrical researches have a physical significance attached to them by the Theory of Screws. This paper also contains the following propositionIf xlt ..., x6 be the co-ordi- nates of a line, and klt ... ke, be constants, then the family of linear complexes denoted by J 2