20 V. Franz Rogel: Darstellung von Strecken und Ebenen. 



Resumé. 



A. Representation of a segment 

 Designe 5 the length of a limited straight line or seg- 

 ment, 5' his projection to the xy - pian and <^5'ír = ^, 



<Í ss' = Vs ^ 5a; = «, cos o? = a, ^ sy = /i, cos i3=^b;<^sz^=y. 

 cos v = c; then is the expression of a segment, that passes 

 the origine 



bl = 5 C (^, ip) 



where C (cp, w) denotes the coefficient ofdirection. 



C ig), v) = (^' COS ifj + ;/ sin \p, 

 where / signifies a bi-compkx or hypercomplex unity, de- 

 finied by 



? = -l 

 theref ore a quantity of two values j = + i ^" = — i. 



Some expressions are deduced for the factor of devi- 

 ation % with which s, must be multiplied in order to recei- 

 ve an other segment s\ The product of two 31 gives an 

 other hypercomplex quantity of the form 



m + in + jr + ijq ^ P where (ij)^ = 1. 



Functions of variables of the form P are of the samé 

 form P. Upon this a deduction is given of factors of de- 

 viation with complex angles «, /^, y. 



The expression for a couple of segments is, — 5) = mx + 

 + imy + jmz' the expression for an arbitr ary 5 (which 

 not passes the origin O: M =(C + ^? 3l)c; 

 the resolution of several geometrical problems. 



B) Representation of a pian; a) which passes the origin O, 



E ^^a-]rih-\- je. 



Composition of two plans, of parallel plans, couple of 

 plans {E, — Eh 

 b) Expression for an arbitrary pian, which not passes O 



- E — a+ib + je + p; 



F^inally several applications. 



