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Evidently a null-system (1,4) can be determined by the equations 
Se + St, + Sit — 0, 
sak == 5, 5% as 5308 — 0. 
Tbe null-points of the line (§) are its points of intersection with 
the curve indicated by the second equation. 
For the curve (P)*+! corresponding to the point P(y) we find, 
by means of the relation 
SY; =f 52s af S342 =; 
the equation 
Pied, ook 
vs 
ak Bk ck | 
zx 
x x 
| 
| 
| 
So the singular points are determined by 
br Di B 
ak bj? 
x zr x 
By harmonic transformation with respect to the conic a, = 0 we 
find a null-system (1, +1), in which the line (9) indicated by aza,=0 
corresponds to the point (7). 
If we put for short 
zeck a, bk = AFH, 2,uk—a,ck = BEH, wv bk ak = CH), 
es ui 5 ae ie ee 1 ae ie 
A 
then we find 
6,18, = AH: BEH: CHH, 
Ans 
1 
ie: 
(a, Ata, Bte, On, +(¢,,A+4,,B +a,,C0)n,+(@,,4+4,,B+a,,C)y,=0, 
and this equation determines with 
EN, HEM, HE = 0 
the new null-system. | 
That it is impossible to deduce any arbitrary (J,4-+ 1) by har- 
monic transformation from null-systems (1, £) can be shown already 
by remarking that the 2(4-+ 1) new singular points furnished by 
this transformation lie on a conic, which does not happen generally 
for k > 2. 
76% 
