NOTES ON TRILINEARS. 537 



In this case, the equation (2) must be satisfied by the value 

 infinity of r, for two distinct sets of values of I, m, n, and these 

 directions are at right angles to each other. Hence 



uP + vm^ + wn^ = 0, 

 for two sets of values of Q, m, n) ; say, {l^, m^, w J, {l^, m^, n^), 

 with the conditions, 



72 + Z2 = 1, m^ + m^ =:l,n + w» =r 1. 

 1 a 1 2 1 a 



Hence 



wZ^ 4- vm^ + wn^ =0, 

 111 



ul^ + vm"^ + vm'' = 0, 

 2 2 a 



and, by addition, we obtain 



M + y + w = 0, 

 which is the condition sought for. 



3. To discriminate the character of the conic (1). 



If the conic be a hyperbola, the two sets of values of (Z, m, n), 

 which make one value of r to be infinite in equation (2) must be 

 real ; if a parabola, they must be equal ; and if an ellipse, they 

 must be imaginary. The condition of a value of r being infinite ia 



ul^ + vm.^ + wn^ = 0. 

 Eliminating n by aid of the relation, 



al + Im -{• en =. 0, 

 we have 



Z2 (m + w — -) 4- m^* (u + M> — ) + 2 ?«z w — = 0, 



and the roots of this quadratic must be real, equal, or imaginary in 

 the three cases. 



Hence for the hyperbola, parabola, or ellipse, respectively, we have 



(^ + ^ ) („ + ^ ) = 0, 



or 



< 



a'^vw + h^wu + c'^wo = 0, 

 > 



4. To find the axes of the conic (1). 



Taking the point (/, y, Ti) for the centre, equation (2) gives for 

 the value of the square of the semi-diameter (r) in any direction, 

 {up + vm^ + v^n^) r^ + vf^ + vg^'+ wTi^ = 0, 



