IV. AXES OF CENTRAL QUADRIC. 569 



We shall show that this always has three real roots. Put 

 A X = u -\- q, 



8) B - I = v -f r, 



C I = w + S, 



where #, r, 5 are to be determined later. Then 



0* 



or arranged according to powers of u, v,'w, 

 uvw 4- qvw + rww + 



9) + u(rs - D 2 ) + v(sq - iJ 2 ) 



+ gr5 + 2DEF - D 2 q - E 2 r - F 2 s = 0. 



Let us now determine q, r, s, so as to make the terms of first order in 

 Uj v, w vanish. 



rs = D 2 , sq = E 2 , qr = F 2 , 



from which by multiplication and division 



10) qrs = 



Thus there remains 



, EF FD . DE 



11J uvw -\ =r- v w H ^- ivu H =- uv = 0. 



Now from 8) 



u = A H q = A EF/D - K = a A, 



12) v = B -X-r = B - FD/E - I = b - A, 



w=C -l- s = C - DE/F - I = c - A, 

 if we write 



A - EF/D = a, B-FDjE=l, C-DE/F=c. 



Also since from 10) #, r, 5 are all of the same sign, let us call them 

 + Z 2 , w 2 , w 2 , so that we have from 11) 



/(T) - (a - A) (6 - A)(c - A) [Z 2 (fc - A) (c - A) 



+ M 2 (c - A) (a - A) + n\a - I) (b - A)J. 

 Substituting for A in turn the values oo, c, ft, a, +00, we obtain 



/(-oo)=oo, 



/>(c) =^ 2 (- C )(&- C ) 

 14) 



/"(a) = 

 /(oo) - - oo. 



