484 
[365 
365. 
ON THE INTERSECTIONS OF A PENCIL OF FOUR LINES 
BY A PENCIL OF TWO LINES. 
[From the Philosophical Magazine, vol. xxix. (1865), pp. 501—503.] 
Plücker has considered (“ Analytisch-geometrische Aphorismen,” Grelle, vol. xi. 
(1834) pp. 26—32) the theory of the eight points which are the intersections of a 
pencil of four lines by any two lines, or say the intersections of a pencil of four lines 
by a pencil of two lines: viz., the eight points may be connected two together by 
twelve new lines; the twelve lines meet two together in forty-two new points; and 
of these, six lie on a line through the centre of the two-line pencil, twelve lie four 
together on three lines through the centre of the four-line pencil, and twenty-four lie 
two together on twelve lines, also through the centre of the four-line pencil. 
The first and third of these theorems, viz. (1) that the six points lie on a line 
through the centre of the two-line pencil, and (3) that the twenty-four points lie two 
together on twelve lines through the centre of the four-line pencil, belong to the 
more simple theory of the intersections of a pencil of three lines by a pencil of tiuo 
lines; the second theorem, viz. (2) the twelve points lie four together on three lines 
through the centre of the four-line pencil, is the only one which properly belongs to 
the theory of the intersections of a pencil of four lines by a pencil of two lines. The 
theorem in question (proved analytically by Plücker) may be proved geometrically by 
means of two fundamental theorems of the geometry of position: these are the 
theorem of two triangles in perspective, and Pascal’s theorem for a line-pair. I proceed 
to show how this is. 
Consider a pencil of two lines meeting a pencil of four lines in the eight points 
(a, b, c, d), (a', b', c'; d'); so that the two lines are abed, a'b'c'd', meeting suppose in