( 750.) 
§ 11. Besides on the @’s the form of an osculating curve depends 
moreover on the 3’s. However for an osculating curve described in 
a definite parallelopiped it depends uot on 3, but only on 2 quantities, 
as is evident when we change the origin of time. We can get: 
Lr == Va, § t » A / a 
n= s COS yk =e (7,8, n,23); ’ 
1 
at, 
y = — os {nt + 2 (n,8, — n,8;)}, 
n 
2 
a 
ue Va, cos (n,-+-n,) t. 
| 
n, Tn, 
a 
The form of the osculating curve evidently depends on the quantities 
B,—8, and 3,—g,. So if we put: 
—-=A,—“=B, Sa C, 2 (n,B,—n,9,)=a, 2 (n,8,—n,8,)—=6, 
then we find 
x = Acos(n,t + a), |} 
y = Beos (nt + b), ere toga ey ee (8) 
z= Ccos(n,+-n,)t. | 
It is evident out of (7) that in the extreme parallelopipeds curves 
are described for which siz y= 0. So for these curves 
2 (n,B, + 2,8, — 7,8, — n,8,) = lx, 
where / is an integer, 
ab Ue 
b= Ila — a. 
So the curves described in the extreme parallelopipeds are given by : 
a == Acos(n,t + a), 
y = B cos (n,t — a + Ia), APR me 
z= C cos(n, + n,)t. 
§ 12. The literature concerning the Lissasous twisted curves seems 
to restrict itself to a paper of A. Rieut (ll Nuovo Cimento, vol IX 
and X, 1873). Rieu discusses only the case that the periods of the 
three mutually perpendicular vibrations have a common measure and 
he investigates which properties of symmetry these curves can have. 
Let us put in (8) /—= Jr and ¢=—r and let us call the values 
of x, y, and z belonging to these values of f, respect. 7,,4,,2, and 
Pas Yo) 2, then we find 
