234 
[36 
36. 
ON THE GEOMETRICAL REPRESENTATION OE THE MOTION 
OF A SOLID BODY. 
[From the Cambridge and Dublin Mathematical Journal, vol. I. (1846), pp. 164—167.] 
Let P, Q, R, be consecutive generating lines of a skew surface, and on these 
take points p', p\ q', q; r', r such that pq', qr' are the shortest distances 
between P and Q, Q and R, &c. Then for the generating line P, the ratio of the 
inclination of the lines P, Q to the distance pq' is said to be “the torsion,” the angle 
q'pq is said to be the deviation, and the ratio of the inclination of the planes Qpq' and 
Qqr' to the inclination of P and Q is said to be the “skew curvature.” And similarly 
for any other generating line; so that the torsion and deviation depend on the position 
of the consecutive line, and the skew curvature on the position of the two consecutive 
lines. The curve pqr is said to be the minimum distance curve [or curve of 
striction]. [When the skew surface degenerates into a developable surface, the torsion 
is infinite, the deviation a right angle, the skew curvature proportional to the curva 
ture of the principal section, i.e. it is the distance of a point from the edge of re 
gression, multiplied into the reciprocal of the radius of curvature, a product which is 
evidently constant along a generating line. Also the curve of minimum distance becomes 
the edge of regression.} A skew surface, considered independently of its position in 
space, is determined when for each generating line we know the torsion, deviation, 
and skew curvature. For, assuming arbitrarily the line P and the point p, also the 
plane in which pq' lies, the position of Q is completely determined from the given 
torsion and deviation; and then Q being known, the position of R is completely de 
termined from the skew curvature for P, and the torsion and deviation for Q; and 
similarly the consecutive generating lines are to be determined. 
Two skew surfaces are said to be “deformations” of each other, when for correspond 
ing generating lines the torsion is always the same. Thus a surface will be deformed if 
considering the elements between the successive generating lines P, Q as rigid, these