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Mathematics. — " The rcprescnlatujn of I he Scrcic-i vf B VLL pns- 

 aiiKj iliroiKjh (t point or hjliuj in a plane, aceonlinfj to the 

 method of ('AroKAM." By I'rof. .1. Cakdinaal. 



1. This comuiuiiication must bo rogarJefl as a contiiuiatioii and 

 onlai'gemeiu uf a lecture delivered at the TO''' Congress of the ''Cie- 

 sellscbaft deutschcr Naturforscher und Acizte" at Dusseldorf (Sept. 

 1898) and published in the last "Jahresbcricht dor Deutschen Ma- 

 Ihematiker-Yoieinigung". There the method applied in the tbllowinu; 

 pages has been considered in its relation to the "Theory of Screws" 

 by Sir R. St. Ball, so I think I can sutfice by beginning with a 

 few brief indications indispensable fur the undcistaiiding uf the pur- 

 pose of the communication. 



2a. The motion of a body considered iiere is tlic motion with 

 freedom of the 4''' degree ; the screws about which motion is possible 

 form a quadratic complex, consisting of all screws reciprocal to a 

 given cylindroid C^. 



b. We construct the screws passing through a point P and be- 

 longing to the complex by drawing perpendiculars through P to 

 the generatrices of C^; each of these perpendiculars moreover inter- 

 sects two generatrices of C^, equidistant from the middle plane (con- 

 jugate lines). The locus of these screws is the cone P". 



('. In a similar manner we construct the screws situated in a 

 plane n. They envelop a parabola Ji~. 



3. The representation of the ra\s of a (juaiiratic complex has been 

 treated among others by R. Stukm and CArOKALl. Wc Knd it inserted 

 at large in Mr. Stdkm's "Liniengeometrie", III, pages '272 - 282. 

 The special complex formed by the screws alluded to belongs to the 

 type treated on pages 438 — 444. Although the results laid down in 

 the following correspond with those obtained there, as could bo 

 expected, there is a great diifereuce in the investigation; tiiis diffe- 

 rence can be circumscribed as follows : 



p' . The proofs are here deduced immediately from the theoi'y of 

 Ball, whereas with Sturm they follow as special cases out of the 

 complex. 



2"'^'. The constructions, more particularly a principal construction, 

 are rtally executed. 



4. Fig. 1 represents the axonometrieal projection uf a cylindroid, 

 whose construction is understood to be known. The nodal line (/ 

 coincides with the axis OZ-^ we suppose further tliat the rotation 



