1365 
ppp gt) 
nrg) Meader ce ene CEU) 
s,=s (p+q—3) 
So we have the theorem : 
The equation (2) represents the osculating plane in the arbitrarily 
chosen point P, (@,,y;, 2) to both the curves Cp, (p, q, s) and Cp, (p,, qu» 5) 
We also find easily for the equation of the osculating plane in 
P, to the curve Cp, (p,, qu 51) 
sh (21) 4h" Li) | tk, ij 
Pi vy 1. YW 8) ay 
so that 
s 
i = Si Th ee iss 
Pi P 
= = P 
and likewise 
OE te Sie GN Ce an Sy 
§ 3. Definition. We call C(p,,q,,8,) the complementary system 
of C(p, q, 5). 
By determining the complementary system of C (Dis Jo 5) we find 
again the original C'(p, q,s), as we have 
Pp, (Poth) =P Pts) (pat) (pt+q—s), 
A (Pt) = 9 (—P+ats) pats) (pt 9), 
8, (pts) = §(— p+g+s) (pts) (p+q—s)- 
Therefore an exception presents itself if and only if we have 
(—p+4at+s) (p—9+s) (p—q+s) = 0. 
For p<.q<s this reduces to the’ possibility p-+q—s=0O; 
on this supposition we find s,=0 and p,=q,, i.e. the system 
C(p‚, qu 8) is the system of the right lines intersecting the axis OZ 
and the line at infinity of the plane z— 0. 
We find s, >0 for sc p+q and s, <0 for s >p+q;p, and 
g, are always positive. 
For p=O we also find p,=0O; then the two complementary 
systems C'(p,q,s) and C'(p,,9,,8,) are both systems of plane curves 
situated in planes « = constant. 
If two of the three numbers p,q, s, e.g. p and q are equal, we 
find p, = q, = ps and both systems C'(p, q,s) and C(p,, q,s s,) con- 
sist in plane curves complanar with the axis OZ. 
The identities 
pP+ qQ + 88 —=0, (12) 
pP+7Q+sS= 0. 
89 
Proceedings Royal Acad. Amsterdam. Vol, XIV. 
