Perpendicularity in a Parallelepipedal System, 21 



should be perpendicular to one another is 



df df df ^ 



or 



df df df ^ 

 aui avi divi 



the condition that the planes of the system 



a o c 

 a e 



(«) 



should be perpendicular to one another is 



or 



d/ii aki all 



and the conditions that the first of the lines (i) should be per- 

 pendicular to the first of the planes (ii) may be written in one 

 or other of the equivalent forms 



(df\ (cl\ /4f\ 



/#\ /#\ 

 \dli) \dk) 



h " k I 



U V 



7. Let us now suppose that the given parallelepipedal sys- 

 tem contains a pair of perpendicular lines (i) ; the condition 

 of perpendicularity gives immediately 



Kuuy + Bvi^i + Ciyt^i + h!{ywx + wvx) + ^'{ii^Ux + xoio) 



+ Q>'(uvi + u^v) - 0. 



Unless, therefore^ the six covariant coefficients are connected hy a 

 linear homogeneous relation having integral coefficients^ no two 

 lines of the system can be perpendicular to one another \ and 

 correlatively, unless the six contravariant coefiicients are con- 

 nected by a similar relation, no two planes of the system can 

 be perpendicular to one another. But the existence of such 

 a relation connecting the six covariant coefficients (or the six 

 contravariant coefficients), though a necessary condition, is 



