﻿312 



Sitzung der physikalisch-mathematischen Klasse 



have therefore an equation == (a' g' li) x* -+- etc., which only 

 differs from the equation T = by having therein the accented 

 letters in place of the unaccented ones: and, substituting for the 

 accented letters their values, the whole divides by the determinant 

 (0qp\|/), and throwing this out we obtain the required equation 

 T=0. 



But it is easier to obtain the equation T = directly : we 

 have 



. hy — gz -f- aw = , 

 - — hx . H- fz -j- hw = , 



• g x —fy • -+- cw = o , 



— ax — hy — er . = , 



viz. in virtue of the equation af -\-hg + ch = which connects 

 the six coordinates, these four equations are equivalent to two in- 

 dependent equations which are the equations of the line (a,&,c,/,#,Ä): 

 or, what is the same thing, any three of these equations imply the 

 fourth equation and also the relation af -f- b g -f- ch = 0. 



We rnight from the three linear relations and any three of 

 the last-mentioned four equations, eliminate a ,b , c ,/ , g , h and so 

 obtain the required equation T = 0; but it is better, introducing 

 the arbitrary coefficients a , ß , 7 , §, to employ all the four equa- 

 tions; the result of the elimination is thus given in the form 



a 



, w 









-z, 



y 



ß 



•> 



w , 





*, 





-X 



7 



•> 





w , 



-y, 



X , 





§ 



, X , 



y ■> 



z , 











A , 



9i > 



Ä M 



a x , 



h. 



Ci 





Ä 1 , 



92 , 



h , 



a 2 , 



K, 



c 2 





fz J 



9s , 



h t 



«3 9 



h , 



Cz 



= 



viz. the left band side here contains the factor — (ccx-hßy-\-xz-\-&w), 

 throwing this out we obtain the required quadric equation T = 0. 



