24 On Perpendicularity in a ParalJelejnpedal System. 

 pression/(.^', y, z) of art. 5 assumes the form 

 f(a;, y, z) = (o^fi + ^2/2 + ^3/3? 

 the ratios of the quantities coi, 0^2, &>3 being irrational, but the 

 coefficients of the quadratic forms fi, /2, /s being integral num- 

 bers. If H(,p, y, z) denote the Jacobian of these three forms, 

 we have the theorem: — - 



'' When the indeterminate equation C = admits of solution, 

 the infinite number of light angles which the system contains 

 all lie on the cubic cone Ji{^'a, yb, zc) = ; viz. an infinite 

 number of lines of the system lie on this cone, and every line 

 of the system w^hich lies on it has a line at right angles to it, 

 also lying on the cone." 



The system may have a simple symmetry or an ellipsoidal 

 symmetry, or none at all ; but it cannot have a spheroidal or a 

 spherical symmetry. 



The conditions for a simple symmetry are that the ternary 

 cubic form C(f, rj, ?) should resolve itself into a rational linear 

 factor and a rational quadratic factor, and that the ternary 

 cubic form H(a',?/,^) should resolve itself into three linear fac- 

 tors. These conditions admit of being further developed (see 

 Dr. Salmon's ' Higher Plane Curves,' pp. 190 and 202 seqg.); 

 it is sufficient for our purpose to observe that the coefficients 

 of the Jacobian H(^', y, ^), no less than those of 0(1", ?;, f), 

 depend solely on the coefficients of the forms yjr^, 1/^2, yjr^, i. e. 

 on the integral numbers entering into the given linear relations. 



The conditions for an ellipsoidal symmetry are that C(f,?7,f) 

 should resolve itself into three rational linear factors, and that 

 H(.r,y, ^j should resolve itself into three factors. 



Two special cases of the general theory (which, however, 

 are not cases of symmetry) deserve attention. 



(1) There may exist in the parallelepipedal system a qua- 

 dratic cone and a plane, such that every line of the system 

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

 it lying in the cone. 



(2) Or, again, the parallelepipedal system may have an in- 

 finite number of pairs of perpendicular lines all lying in the 

 same plane ; and it may also have at the same time a second 

 set of such pairs lying on the surface of a quadratic cone, the 

 plane of each pair of this second set passing through the polar 

 line of the first-named pair with regard to the cone. 



11. Case of four linear relations heticeen the coefficients. 



Here every line, without exception, of the parallelepipedal 

 system has a line at right angles to it ; and this distribution 

 of pairs of perpendicular lines may exist without the presence 

 of any symmetry whatever. The symmetry (if any) may be 

 simple, or ellipsoidal, or spheroidal, but cannot be spherical. 



