61] 
ON GEOMETRICAL RECIPROCITY. 
381 
then if the position of a point be determined by the coordinates a, ¡3, y, the equation 
of one of the corresponding lines is 
a ^A/3y + 7^=0, 
(that of the other is obtainable from this by writing a, b. c; a', b', c'; a", b", c", for 
a, a', a"; b, b', b"; c, c', c"). Hence if the point lies in the corresponding line, this 
equation must be satisfied by putting a, /3, 7 for x, y, z\ or, substituting x, y, z 
in the place of a, ¡3, 7, the point must lie in the conic 
U = ax 2 + b'y 2 + c"z 2 + (b" A c') yzA(c A a") zx A (a' + b)xy = 0, 
(which equation is evidently not altered by interchanging the coefficients, as above). 
Again, determining the curve traced out by the line a%+/3y + 7^=0, when a, /3, 7 
are connected by the equation into which U=0 is transformed by the substitution of 
these letters for x, y, z; we obtain 
1 , V » 
2 a , 
a' A b, 
a" Ac 
V> 
a A b, 
2b' , 
b"Ac' 
l 
a Ac, 
Vac', 
2c" 
which is also a conic. It only remains to be seen that the conics U = 0, V = 0 have 
a double contact. Writing for shortness 
V = 
a , 
b , 
b f , 
b", 
c 
c' 
c 
// 
it may be seen by expansion that the following equation is identically true, 
V = 4V U - [x (ab" - a"b + ac - ac') + y (b'c - be' + b"a - b'a") + s (cV - c'a" + cb" - c"b)Y, 
which proves the property in question. 
Suppose the equations of the two conics to be given, and let it be required to 
determine the corresponding lines to the point defined by the coordinates a, /3, 7. 
Writing, to abbreviate, 
' U = Ax 2 + By 2 + Gz 2 + 2Fyz + 2Gzx + 3.Hxy, 
U 0 = Aol 2 + B/3 2 + Cy 2 + 2F/3y + 2 Gy a + 2Hal3, 
W = Aax + B/3y + Oyz + F (/3z + 7y) + G (yx + az) + H (ay + j3x), 
P = lx + my + nz, 
P 0 = la + m(3 + ny, 
k K = ABC - AF 2 -BG 2 - CH 2 + 2FGH,