308 G. H. KNIBBS. 



the result the conoidal form EOE', what were straight lines 

 originally being now curved inwards, and the characteristic features 

 of elliptic space would have been developed. 1 Tortuous seolotropic 

 space may be reduced to isotropy by expansions, contractions, and 

 rotations, linear motion being included in the last. 



33. Conformal representation of functional dependence. — The 

 part played in the modern theory of functions by the numerically 

 impossible quantity V-l, called therefore imaginary, is of such 

 moment, that a consideration of the principle of continuity in the 

 development of geometrical figures, necessitates at least a brief 

 reference thereto. 



Let for brevity this quantity be denoted by i, and the complex 

 quantity (x + iy) be denoted by z, the part x being real, and the 

 part iy, y times the imaginary i<. We have seen that imaginary 

 quantities can be represented upon an infinite lemniscate cylinder 2 

 when z = iy only, z being a line. Suppose z however to denote 

 merely the place of a point ; this can be represented among other 

 ways by "Argand's diagram," 3 Fig. 30, or by Neumann's sphere: 4 



x is the distance Ox, y the distance 

 xz ; and obviously 



p cos B = x; p sin = iy (33) 



hence 



; = o(cos# -f i sinO) = p cis# = pe ie . . . (34) 

 the last two being merely abbrevia- 

 tions of the first expression in (34). 

 Suppose further, w; to be a point 

 dependent upon z, (regarded as an 

 irresoluble quantity) in such a way that a single series of opera- 



1 See the left hand side of Fig. 20. 



2 See Fig. 4, § 13, formula (14), and footnotes 1 - 4, p. 246 and 1 p. 247. 



3 See footnote 4 page 246. Kuhn's name ought to be associated with 

 this diagram. 



4 See his Vorlesungen iiber Biemann's Theorie der Abelschen Integrale 

 Leipzig, Teubner, 2° Edit. 1884. A sphere of unit diameter, was chosen 

 by Neumann as the field or surface on which to represent the position of 

 p. For an infinite or a large value of z, this method has advantages, but 

 it is unessential to our present purpose. 





CQ 























M 











ea 







z 





>» 











u 













cS 













'Si; 



p 











a 









5*5 



•<s> 





t— i 

















6 





X 









Eeal 



axis. 









Fig. 



30. 







