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We call the base points of (3) S,,, Sas S,,, S,,, the join of 
S,, and S,, is called s,,, that of S,, and S,, bes,,..To the pencil (3) 
belongs a conic which has degenerated into (s,,, s,,); let z be inter- 
sected by the first of these lines in A,,, by the second in A,,; the 
point of z harmonically separated from A,, by &, is B,,; also A, 
and B,, are harmonically separated by 8. The conic (pair of lines) 
of (4) corresponding to this degenerate conic of (3), is therefore 
formed by OB,, and OB,,. Let the intersection of OB,, with s,, 
be called §S,,, that with s,, be 7,,; S,, and 7’, are points of y,. 
Now O is the pole of z with regard to 8; OB,, is therefore the 
polar line of A,,; for this reason the pole of s,, lies on OB. 
We knew already that the latter point also lies on s,,; hence S,, is 
the pole of s,, with regard to 3. In this way we find that the pole 
of each side of the quadrangle S,,S,,S,,.S,, lies on y,. Now the 
four corners of a polar quadrangle form with the six poles of the 
sides a configuration (10,, 10,) of Drsaraums; hence all the corners 
of this configuration lie on y,. 
It is noteworthy that the points Sj;, Sm Sri always lie on a line 
Sim Of which Sj, is the pole. Each of these lines has a fourth point 
of intersection with y,; if we choose e.g. s,,, it cuts y, besides in 
Ss Sus Soo in one more point 7’, which also lies on OS. 
By giving to the equation of y, the form 
¥,=2eyp + {(2—O) 27+ (14 0O)y7}w=0-... (5) 
we have been able to show that in it a Cf. (10,,10,) of Desarcuns 
is inscribed. 
But the equation (5) also contains the entirely arbitrary constant 
7; by varying this we shall find an infinity of pencils (3) and (4) 
and an infinity of configurations. Hence: 
In y an infinity of configurations (10,,10,) of Desareuus can be 
inscribed ; each configuration is self-polar with regard to 3. 
5. In (5) p and w are functions of x,y,z and C. Let P(a',y/, z’) 
be an arbitrary point of y,, so that 
VAN (o Wine) = 0; 
Let us then determine C, so that 
ODE a OSU 
We can find two values of C’ satisfying this condition; then also 
according to (5) 
OE a5 AN = de 
If we therefore consider C as a variable parameter, each point of 
y is twice a base point of a pencil (3). On the other hand it is 
