APPENDIX I. 185 



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 be- 

 schranken : es treten geometrische Gebilde auf, welche zu Dyna- 

 men in derselben Beziehung stehen, wie gerade Linien zu 

 Kraften und Rotationen." There can be little doubt that the 

 " geometrische Gebilde," to which Pliicker refers, are what we 

 have called screws. 



As we have already stated ( 16), it is in this book that we 

 find the first mention of the surface which we call the cylindroid. 



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 sur- 

 face the cone breaks up into two planes. This surface is of the 

 fourth class and fourth degree, and is known as Rummer's sur- 

 face, or as the surface of singularities appropriate to the given 

 linear complex. (See Kummer, Abhandl. der Berl. Akad., 1866). 

 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. 



KLEIN (Felix). Zur Theorie der Linien- Complex e des ersten und 

 zweiten Grades. Math. Ann., II. Band, pp. 198-226 

 ( i4th June, 1869). 



The "simultaneous invariant" of two linear complexes is 

 discussed. In our language this function is the virtual co- 

 efficient 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 re- 

 searches have a physical significance attached to them by the 

 Theory of Screws. 



This paper also contains the following proposition: If 

 ,* . . ., x t be the co-ordinates of a line, and k vt ... k t be con- 



