110 UNIVERSITY OF COLOR ADO STUDIES 



jpla/nes of the tetrahedron. T%e diagonals of the six plane quadri- 

 laterals formed hy the eight points form three closed ohlique quadri- 

 laterals whose vertices are on the edges of the tetrahedron^ and 

 through each of these quadrilaterals passes a hyperholoid belonging 

 to the pencil through the C^. 



Considering the projection of the whole configuration, Fig. 4 

 shows immediately that BCDA* and B*C*D*A are quadruples on 

 the C„ since they are the intersections of the pencils A. MiMjMjM^ 

 and A.*MjM2M3M^ with the C3. This proves Clebsch's first theorem. 

 The second theorem appears from the fact that the sixteen lines join- 

 ing the points of the quadruples BCDA* and B*C*D*A intersect 

 each other in the quadruple MjM.^MjM^. In the figure the point of 

 intersection K of MjM., and MjM^ is the trace of a ray through the 

 center of projection C and cutting M2M3 and MjM^; this ray meets 

 the C in another point and its trace, being the projection of this 

 point, necessarily belongs to the C^. The same is true of the remain- 

 ing diagonal points of the quadruple MiM2M3M^. 



5. In preparing Fig. 4 I have shown only the most important 

 parts of the construction. For a detailed descriptive-pro jective 

 execution I refer to Fiedler, loc. cit. 



It goes without saying that by the same method, employing 

 higher involutions, closed polygons of 2n sides may be obtained. 

 For the purpose of this paper it is suflicient to establish by this 

 method some of the propositions of §8. 



The complete geometrical theory of Steiner's polygons is given 

 in Disteli's work, loc. cit. 



III. APPLICATION OF ELLIPTIC FUNCTIONS TO CERTAIN LINKAGES. 



§10. Peaucellier's Inversor. 



1. The element of the linkage which I shall consider consists 

 of a Peaucellier's Cell or Inversor^. 



(') A description of this particular link-work may be found in every modem text-book on 

 kinematics. The constructions that it performs were proposed by Peaucellier in the 

 Nouvelles Annates de Mathematiques, ser. 2, vol. 3 '1864). p. 414, and the link-work was 

 published by him in the same journal, ser. 2, vol. 12 '1873), p. 71. 



An account of the application of elliptic functions to Peaucellier's inversor appeared in the 

 Annals of Mathematics, 2nd series, Vol. 2, No. 2. 



