486 INTERSECTIONS OF A PENCIL OF FOUR LINES BY A PENCIL OF TWO LINES. [365 
It remains to be shown that the two lines through P, viz. the line containing 
ab'. dc and a'b. d'c, and the line containing ad'. be and a'd. cb', are one and the same 
line. This will be the case if, for instance, ab'. dc' and ad'. be' also lie in a line 
through P. 
C' 
We have the points (a, b, d) in a line, and the points (b', c', d') in a line; the 
points a, d, b', c are also called A, B', B, A' respectively; ad', bb' meet in G, and 
be', dd' meet in G'; hence, considering the hexagon ad'db’bc', the lines 
ad', b'b meet in C , 
d'd, be' „ C' , 
db', ca' „ AA'. BB'; 
and hence these three points lie in a line; or, what is the same thing, the lines 
AA', BB', and GC' meet in a point; that is, the triangles ABG, A'B'G' are in 
perspective: the corresponding sides are 
AB, A'B', that is, ab', c'd, meeting in ab'. c'd, 
BG, B'G' „ b'b, d'd, „ P , 
GA, C'A' „ ad, be', „ ad', be'; 
and these three points lie in a line; that is, the points ab'. dc' and ad'. be' lie in a 
line through P. Hence the line through ab'. dc and a'b. d'c and the line through 
ad'. be' and a'd. cb' are one and the same line; that is, 
the points ab'. dc, a'b. d'c, ad . be', a'd. b'c lie in a line through P. 
This proves the existence of one of the lines through P; and that of the other two 
lines follows from the symmetry of the figure; it thus appears that the twelve points 
lie four together on three lines through P. 
Cambridge, April 11, 1865.