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prime factors only occurring only once in each expression; then we 
must have é 
LMN:0? = —(ZL4+M+4N):8?. . . s . (@1) 
By substituting the value of —(L + M+ N) following from it 
into the expression for p:q given above we easily find: 
MEE EE Ep ee 
M(atbN) —a(L+N) _ M(arbD) 
So, as soon as L, M, N satisfy the condition (21) we find sets 
of numbers (p,q, 8), (p', q', s') and therefore also two sets of curves 
C(p,q.s), Cp, 7, s') lying on the surface (18). After some reductions 
we find p':q':s'=p,:9,:58, as the deduction of (p, q, s) and (p’, q’, 5) 
requires. So we have proved the theorem: 
A surface 
elyMzN — B 
admits as asymptotic curves the curves of the systems 
C (M (a+)N), — a(L4-N), M (a—bD)), 
and 
C (M (a—bN), — a(L4+N), M (a+b), 
as soon as L, M, N satisfy the condition 
b? 
L+MiNn=—— LUN, 
a 
a and b being integers. 
The simplest example of a surface zl yMz2N — B, where the con- 
b? : 
dition L+ M+ N=— LMN, holds, is the hyperbolic paraboloid 
a 
a2 == Bye 
In this case the equations (22) become 
p= 0, gs sand sr =O D= ga 
The systems C(O,1,1) and C'(1,1,0) are systems of right lines 
forming on the paraboloid the asymptotic lines. 
§ 6. Any surface O,,, contains besides the two systems of asymp- 
totic lines C(p, q, s) and C(p,, q,, 5,) other systems of curves belonging 
to the systems 
C(p + Arig + agus + As,) 
“and this holds for any rational value of À either positive or negative. 
Let, in order to show this, P, (x,,3',, 2,) be any point of Oren 
that we have 
a, Pkz Sk — By, —, 
then O.-, contains any point of the curve 
