( 211 ) 
1°. If we state that the five lines among which the relation of the 
five lines @ exists, are conjugate to each other, then each row or 
each column of the determinant contains five conjugate lines. 
2°, Hach of these six quintuples of conjugate lines leads back 
to the fifteen lines, if we search for the lines of intersection of the 
ten spaces through the lines of the quintuples two by two. 
30, Every two rows or two columns of the determinant furnish 
a double four after omission of the two elements of which the 
corresponding ones are wanting. 
The proof for the first law is immediately given. If we connect 
with the first row one of the others, say the fourth, it follows from 
the definition of the last merely, that 
is a double four and that the four spaces (4, a), (dy aa), (cs a4), (bg as) 
intersect each other according to a right line, which must be dg, 
the five lines a being conjugate; so the five lines of the fourth row 
are also conjugate. 
The second law is immediately proved out of the second of the 
two tables. And the third follows out of the first table, when in 
conneetion with the above-named determinant we reduce this to the 
observation that two of the fifteen lines intersect each other when 
they belong nowhere in the determinant to the same row or the 
same column. The word “nowhere” inserted here refers to the cir- 
cumstance, that each line appears twice. So taken all together the 
system of the fifteen lines contains fifteen double fours; each of these 
we can call in the configuration opposite to the right line, which 
is the section of the four spaces passing through the opposite elements 
of the double four. 
The analytical representation of the fifteen right lines as well as 
the notation of the determinant leaves still something to be desired. 
In one as well as the other the circumstance that four of the fifteen 
lines a are brought to the foreground harms the regularity. Firstly 
we shall now try to improve the notation of the determinant. 
Starting from the five lines a the other ten lines are found as 
common transversals of the triplets to be formed out of those five 
lines. This leads to the idea of representing all lines by a, where 
the original lines a retain their index for the present, each of the 
14 
Proceedings Royal Acad. Amsterdam Vol. IV. 
