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§ 26. Osculating curves. The image point which is to represent 
the movement of the mechanism with 4 degrees of freedom, moves 
in a space Zi. The coordinates w, y, z, and « of the image point 
on a rectangular system of coordinates are at every moment equal to 
Gi» Jo Jao Yq. Its movement is then determined bv: 
Va, 
a= — COS (n,tt-2n, B.) ; 
ne 
Va, a 
Y = — cos (n,t+-2n, B) , 
ie 
a, ) 
RI cos (nst +-2n, B;) , 
ns 
Va 
= 2 ane Si : ì 
U —-—— — cos (rn, +n, En,)t + 2 (n, Sn, dn,) B} 
AD, EN, 
If we ascribe to the es and @'s their momentary values, then 
these equations represent the osculating curves for the indicated 
moment. The osculating curve we can call a Lissasous curve. 
The curve remains enclosed inside a fourdimensional parallelotope 
bounded by the spaces: 
fr cis Ye ees ox ee ge a= Lass i. at eN 
n, Ns nn, En, 
By the variability of the «’s the ecireumseribed parallelotope also 
changes; the vertices move backward and forward between two 
extreme positions along a wrung curve; this curve lies on a hyperel- 
lipsoid, whose axes lying along the axes of coordinates are proportional to 
een! CP f 
n 
aarp) : 
, A a EAO 
The form of the wrung Lissasous curve in a definite parallelotope 
depends, as is found by a change of the origin of time, on the 
quantities 
2n, (BP): 2n, (Sel 2n, (B, zig) 
The osculating curves described in the extreme parallelotopes have 
the property that 
an, (6,—8,) + 2n, (°,—8,) = 2n, (BB) = lm. 
For these curves the relation holds: 
a y z u 
cos—! — + cos—! = (naa! = = cos! —, 
A B C D 
when A, B, C, and P are written for the amplitudes and 7 is 
supposed even. 
