Perpendicular iti/ in a Parallelepipedal System. 23 



and by 6, 6\ 6" tlie roots of the cliscriininantiil cubic oC 

 ■^1 + ^^2- I^' tliese roots are irrational, the system contains 

 not a single pair of pei'pendicular lines. If one of them, for 

 example 6, is rational, we still have to examine whether the 

 factors of y^i + O-^^ ai'e rational ; if they are, we have a pair 

 of perpendicular lines. If all the three roots 6, Q' ^ Q" are 

 rational, we have to examine the ftictors of each of the three 

 forms ^i + ^'^2j '^i + ^^^2j '^i + ^^^i^2) according as these 

 factors are or are not rational (if the factors of two of them 

 are rational the factors of the third are so too), we obtain one 

 or three pairs of perpendicular lines, or no pair at all of such 

 lines. 



When two of the roots 6, 6' , 0" are equal, we have either 

 one, and only one, pair of perpendicular lines ; or we may have 

 tu'o pairs, the plane of one of the right angles containing one 

 of the rays of the other right angle. When the three roots are 

 all equal we have a single pair of perpendicular lines. 



Lastly, the coefficients of the discriminating cubic may all 

 vanish. If this happens, either (a) "^^ and ^2 differ, if at all, 

 by a numerical factor, and every line of the system that lies 

 in a certain plane has a line of the system at right angles to it 

 in the same plane ; or (/3) -y^i and -^2 have a common linear 

 factor, and the system possesses a simple symmetry. 



We may thus enunciate the theorem: — 



'' The conditions that a parallelepipedal system should pos- 

 sess a simple symmetry are (a) that the coefficients should be 

 connected by two linear relations, (h) that the two quadratic 

 forms appertaining to these relations should have a linear fac- 

 tor in common." 



10. Case of three linear relations hetween the coefficients. 



We represent by -^i, -^1^2, ^^3 the quadratic forms appertain- 

 ing to the given relations, and we obtain the following 

 theorem : — 



'' The system contains no right angle, or an infinite num« 

 ber, according as the indeterminate cubic equation 



C = 



d^2 

 df 



dr) 



'djr^^ 

 dr) 



d^^ 

 drj 



d±i 

 d^ 



d±2 



cK 



d±l 



dt 



= 



does or does not admit of solution in integral numbers," 

 By virtue of the three given relations the characteristic ex- 



