928] 
ON THE ANALYTICAL THEORY OF THE CONGRUENCY. 
229 
viz. the consecutive line (a 1 , b 1} c 1( f 1} g 1} h x ) belongs to the linear congruency defined 
by these two equations. 
Forming with these a linear combination 
(XF gF) Uj + (XG + gG') &i+ (XH + gH'') Ci + (Ad. + gA /s ) f l + (XB + f xB')g 1 4 (XG 4* gG) h x = 0, 
we may determine the ratio X : g by the equation 
(Ad + gA’) (XF + gF') 4- (XB + gB') (XG + gG') + (XG + gG') (XH + gH') = 0, 
that is, 
A (AF + BG + CH) + Xg (AF' + BG' + GH' + FA' + GB' + HC')+g* (A'F'+B'G' + G'H') = 0 ; 
we have thus two values of X : g; and, denoting the corresponding values of 
XF+gF', ..., XC + gC' by (/ 2 , g 2 , h 2 , a 2 , b 2 , c 2 ) and (/ 3 , g 3 , h 3 , a 3 , b 3 , c 3 ) respectively, 
we have 
«2/2 + b,g, 2 + c 2 h 2 = 0, a 3 f 3 + b 3 g 3 + c 3 h 3 = 0, 
and 
«1/2 + b x g 2 + cju + f x a 2 + g 1 6 2 + h x c, = 0, 
«1/3 + 61g-i + cA +f x a 3 + gj)3 4- Kc 3 = 0; 
viz. we have thus two lines (a 2 , b 2 , c 2 , f 2 , g 2) h 2 ), (a s , b s , c 3 , f 3 , g 3 , h 3 ), not in 
general meeting each other, each of which is met by the line (a l5 b 1} c x , f u g 1} 7г 1 ); 
say, for shortness, the lines (a, b, c, f, g, h), (u l5 b lt c 1} f 1} g x , K), (a 2 , b 2 , c 2 , f 2 , g 2 , h 2 ), 
(a 3 , b 3 , c 3 , /3, g 3 , h 3 ) are the lines 0, 1, 2, 3 respectively. 
We may, in the foregoing investigation, substitute, for the coordinates of the 
line 1, those of the line 0; and it hence appears—what is indeed obvious—that the 
line 0 meets each of the lines 2 and 3. Supposing now that the lines 0 and 1 
meet each other, that is, that we have 
q/1 + bg 1 + cJh 4- f(h + gb x 4- hc x = 0, 
then it is clear that the line 1 must pass through the intersection of the lines 0, 
2, or else through the intersection of the lines 0, 3; in fact, if 0 and 1 intersect 
in a point not on the line 2 or 3, then we have the line 0 as a line passing 
through this point and meeting each of the lines 2 and 3; and also the line 1 as 
a line passing through this point and meeting each of the lines 2 and 3; that is, 
the lines 0 and 1 would be one and the same line. 
It thus appears that, considering the line 0 as given, we have two lines 2 and 
3 each meeting this line, say in the points P 2 and P 3 respectively; and that, this 
being so, the consecutive line 1 meets the line 0 either in the point P 2 or else in 
the point P 3 , viz. that there are two consecutive lines 1, say 1 2 and 1 3 , meeting 
the line 0 in the points 2 and 3 respectively. These points are thus given as the 
intersections of the line (a, b, c, f g, h) with the lines (a 2 , b 2 , c 2 , f 2 , g 2 , h 2 ), 
(a 3 , b 3 , c 3 , f 3 , g 3 , h 3 ) respectively; viz. supposing that X : g is determined by the 
above-mentioned quadric equation, and calling its roots \ 2 : g 2 and X 3 : g 3 , then we 
have 
a 2 , b 2 , c 2 , f 2 , g 2 , h 2 = X 2 ^l 4 g 2 A , ..., X 2 H4" g%H , 
a 3 , b 3 , c 3 , f 3 , g 3 , h 3 — Y3.A + g 3 A , ..., X 3 H 4* g 3 H.