GENERATION OF FIGURES ON STRAIGHT OR TORTUOUS AXES. 319 



theory of curved space, are really conquests in this field, and only 

 require to be dissociated from the space-theory to be seen as such. 

 Non-euclidean as a descriptive adjective hardly conveys the right 

 idea of this higher general geometry; while the term "Pan- 

 geometry" does define its essential character. This geometry is 

 not opposed to euclidean, but is a supplement of its field, the 

 the narrowness of which has been exposed by the magnificent 

 researches of modern mathematicians. 



Some THEOREMS concerning GEOMETRICAL FIGURES 



IN SPACE OF tt-DIMENSIONS, WHOSE {n - I) DIMENSIONAL 

 GENERATRICES ARE n ic FUNCTIONS OF THEIR POSITION ON AN 

 AXIS, STRAIGHT, CURVED OR TORTUOUS. 



By G. H. Knibbs, f.r.a.s., 

 Lecturer in Surveying, University of Sydney. 



[Read before the Royal Society of N. S. Wales, December 4, 1901.'] 



1. Problem defined. 



2. Form of graph of generating function immaterial. 



3. Oblique rectilinear ages. 



4. Generating function with curvilinear axes. 



5. Rotation of generatrix obliquely about the t-axis. 



6. Curvature of the principal, i.e. the £-axis. 



7. Tortuosity of the principal axis. 



8. Theory of equivalent generatrix of unit value. 



9. Necessary number of equidistant values of equivalent generatrix. 

 10. Conclusion. 



1. Problem defined. 1 — Whenever a finite quantity V t in w-dimen- 

 sional homaloidal space, generated by the motion of an (n-1)- 



1 The problem discussed may be read as a continuation of a previous 

 paper, entitled, " On the relation, in determining the volume of solids, 

 whose parallel transverse sections are n lc functions of their positions on 

 the axis, between the number, position, and coefficients of the sections, 

 and the (positive) indices of the functions." See Journ. Roy. Soc, N.S. 

 Wales, Vol. xxxiv., pp. 36-71. The z axis in that paper is for obvious 

 _reasoDS denoted by t in this. 



