of a Surface of the Second Order into itself 331 



the discriminant vanishes ; we have for the determination of q a 

 quadratic equation, which may be written 

 2 2 +(2 + /% + K=0. 



Let the roots of this equation be q /} q u ; then each of the func- 

 tions ^U + V, q tl U + V will break up into linear factors, and we 

 may write 



?,U + V=R / S / , 



g„U + V=RA. 



U and V are of course linear functions of R ) S / and R^S,,, the 

 form which puts in evidence the fact of the two surfaces inter- 

 secting in four lines. 

 The equations 



# 1 +a? 2 =2£ Pi + Pq=^V, ^1 + ^2=% w 1 + w <2 =2co, 

 show that the point (f, 77, £ co) lies in the line joining the points 

 ( x i> V\i 2 \> w \) an d (# 2 , y& Z& w<z) ', and to show that this line 

 touches the surface V=0, it is only necessary to form the equa- 

 tion of the tangent plane at the point (f, t], f, co) of the surface 

 in question ; this is 



{oc + vy— fjLZ + aw)(l; + vn— /*£+«») + .. =0; 

 or what is the same thing, 



(x + vy— fiz + aw)x l + . . =0, 



which is satisfied by writing x v y v z v w l for (x, y, z } w) } i. e. 

 the tangent plane of the surface contains the point (x v y li z v w l ). 

 We see, therefore, that the line through (x v y v z v w^) and 

 fe y& *& w 2 ) touches the surface V=0 at the point (f, 77, £ &>). 

 Write now 



a'-— ti-t£ <J-— X'- — ///- — 1/ = — 

 4>' </>' T' T' ^ *' *' 



If we derive from the coordinates x v y v z v w v by means of 

 the coefficients a', b\ c\ V, /jj, /, new coordinates in the same 

 way as x 2 , y 2) z 2 , w 2 were derived by means of the coefficients 

 0,&,c,X,/u,,v, the coordinates so obtained are — a? 2 , — y v —z 2 ,—w 2 , 

 i. e. we obtain the very same point (# 2 , y^, z^ w 2 ) by means 

 of the coefficients {a, b, c, X, //>, v), and by means of the coeffi- 

 cients («', b\ c', X', /J, v ! ). Call P, rj 3 p co 1 what f, 17, g g> 

 become when the second system of coefficients is substituted for 

 the first ; the point £', 77', f ', co 1 will be a point on the surface 

 V f =0, where 

 V'=cj>%x2 + y2 + z* + wZ) + (-cy + bz-\w) 2 + (--az + cx-fj,w) 2 



+ ( — bx + ay — vw) 2 + ( — X#— py — vz) q . 

 And since 



V + V'ss^ + jf + ^ + w 2 ), 

 Z2 



