BIBLIOGRAPHICAL NOTES. 517 



When the forces are fewer than seven, the formula? 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 OJ P lt 

 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 formulae 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. 



PLUCKER (J.) Neue Geometrie des Raumes gegrundet auf die Betrachtung der 

 geraden Linie als Raumelement. Leipzig (B. G. Teiibner, 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 &quot;Dyname&quot;: &quot; 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 iiberhaupt, unserem Erkennungsvermogeii entzieht, die Bewegung selbst, 

 statt der Ursache die Wirkung, bezeichnet.&quot; 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 : &quot;Dann entschwindet das specifisch Mechanische, und, um 

 mich auf eine kurze Andeutung zu beschranken : es treten geometrische Gebilde 

 auf, welche zu Dynamen in derselben Beziehung stehen, wie gerade Linien zu 

 Kraften und Rotationen.&quot; There can be little doubt that the &quot; geometrische 

 Gebilde,&quot; to which Pliicker 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. n., pp. 198-226 (14th June, 1869). 



The &quot; simultaneous invariant &quot; 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 proposition: If x l , ..., x 6 be the co-ordi 

 nates of a line, and & n ... k 6 , be constants, then the family of linear complexes 

 denoted by 



a, 1 ! 2 x 6 2 



/Cj A, A/g A. 



