36] ON THE GEOMETRICAL REPRESENTATION OF THE MOTION OF A SOLID BODY. 235 
elements be made to revolve round the successive generating lines P, Q and to 
slide along them. {They are “ transformations”, when not only the torsions but also the 
deviations are equal at corresponding generating lines: thus, if the sliding of the 
elements along P, Q be omitted, the new surface will be, not a deformation, but 
a transformation of the other.} No two skew surfaces can be made to roll and slide 
one upon the other, so that their successive generating lines coincide, unless one of 
them is a deformation of the other: and when this is the case, the rolling and sliding 
motions are completely determined. In fact the angular velocity of the generating line 
is the angular velocity round this line, into the difference of the skew curvatures of 
the two surfaces; the velocity of translation of the generating line in its own direction 
is to the angular velocity of the generating line, as the difference of- the deviations 
is to the torsion. {This includes also the case in which one surface is a transforma 
tion of the other, where the motion is evidently a rolling one.} A skew surface moving 
in this manner upon another of which it is the deformation, may be said to “ glide ” 
upon it. We may now state the kinematical theorem: 
“ Any motion whatever of a solid body in space may be represented as the ‘ gliding 
motion of one skew surface upon another fixed in space, and of which it is the defor 
mation.” 
a theorem which is to be considered as the generalization of the well-known one— 
“Any motion of a solid body round a fixed point may be represented as the rolling 
motion of a conical surface upon a second conical surface fixed in space.” 
and of the supplementary theorem— 
“ The angular velocity round the line of contact (the instantaneous axis) is to the 
angular velocity of this line as the difference of curvatures of the two cones at any 
point in the same line, to the reciprocal of the distance of the point from the vertex.” 
The analytical demonstration of this last theorem is rather interesting: it depends 
on the following formulae. Forming two determinants, the first with the angular velo 
cities round three axes fixed in space, and the first and second derived coefficients 
of these velocities with respect to the time; the other in the same way with the 
angular velocities round axes fixed in the body; the difference of these determinants 
is equal to the fourth power of the angular velocity into the square of the angular 
velocity of the instantaneous axis. 
To show this, let p, q, r be the angular velocities round the axes fixed in the body; 
u, v, w those round axes fixed in space; co the angular velocity round the instantaneous 
axis; V, il the two determinants: the theorem comes to 
V - n = M, 
where M = co 2 (pi' 2 + q' 2 + r' 2 — co' 2 ), or co 2 (u' 2 + v' 2 + w' 2 — co' 2 ). 
Here 
u = a p + /3 q +7 r , 
v = a! p + /3' q +7' r , 
w = a"p + (3"q + 7"r; 
30—2