1363 
Mathematics. — “On a class of surfaces with alyebraic asymptotic 
By Prof. W. A. VersLuys. (Communicated by Prof. 
bd 
curves.’ 
J. CARDINAAL). 
PQs 
§ 1. Let a twisted curve cl ) be given by tbe equations: 
Re 
want iedee jn nt Te oe ar oe 
t being the arbitrary parameter, a, 6,¢ constants and p,g,s positive 
integers not admitting a common divisor In general we suppose 
pq<s. 
By assigning to a,b,c all possible values we get a system of . 
oo? curves, which will be denoted as the system C(p,q,s). All 
the curves of this system contain the origin 0 and the point at 
infinity on the axis OZ; through any other point of space only one 
curve of the system C(p,q,s) passes. The curve determined by the 
point A shall be indicated by C4(p,q,s) or by Cy. 
Pr Qs 
Let P,(x,,y,,2,) be the point of the curve cf ) corresponding 
a, O, C 
to the value ¢, of the parameter ¢; then 
BdP ver Hitt Ln Ares EE. 
The equation of the osculating plane in P, to Cp, is: 
| x — Hu, YY, zz, | 
| path! qbtal ects! | 
| | 
p(p—L at? g(q—l) by? se iets 
or reduced 
| 2, | 
| Pz, Py, 28. | ==), 
| pte, "4; sz, 
or worked out 
P wv, q Yi € ey 
By putting 
ne ep 4 Pero ; ee PASS) eee meee, | 
p q 8 
and replacing P+ Q+ 8S by the value — PQS, equal to it, the 
equation of the osculating plane becomes 
ER 
Y 
1 
Er OEE 
2 
1 
P—+Q 
v, 
