PKINCIPLE OF CONTINUITY IN THE THEOEY OF SPACE. 243 



On the principle of CONTINUITY in the GENERATION 



of GEOMETRICAL FIGURES in PURE and PSEUDO- 



HOMALOIDAL SPACE of ^-DIMENSIONS. 1 



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.'] 



0. Introductory. 



1. The dimensionless point as generatrix. 



2. Summational generation impossible with dimensionless point. 



3. The straight line. 



4. Fluxional generation by means of dimensionless point. 



5. General laws of fluxional generation. 



6. Inverse fluxional generation and its laws. 



7. Zeros and infinities of u-dimensions. 



8. Summational generation by means of n-dioiensional point. 



9. Zeros and infinities of successive orders. 



10. Spatial continuity and its numerical expression. 



11. Rotational generation. 



12. Finitely and absolutely homaloidal 2-dimensional space. 



13. Resolution of discontinuity in 2-dimensional curves through 



infinite paths in 3-dimensional space. 



14. Infinitesimal approximation and absolute identity in differential 



coefficients. 



15. The theory of metrics. 



16. The projective theory of distance. 



17. The theory of linear intensity. 



18. Space of non-uniform intensity. 



19. Complex space. 



20. Space of positive and negative curvature. 



21. Geometrical illustration of elliptic and hyperbolic space. 



22. Symmetrical, elliptic and hyperbolic space of two-dimensions. 



23. Impossibility of elliptic or hyperbolic space existing in a pure 



homaloid of the same number of dimensions. 



1 The term dimension is used in the sense that a line is essentially of 

 unit dimension, a surface 2-dimensional, a volume 3-dimensional, and. so 

 on. The space of points and planes, is sometimes said to be of three 

 dimensions ; but of lines, of four dimensions, e.g., as by Henrici. 



