236 
Proceedings of the Royal IrUh Acade/ni/. 
atout which the body is free to twisty there is a single screw X which 
is reciprocal to A^, &c., A^. Every screw co-ordiuate with A^, &c., 
must be reciprocal to X Thus we may choose any four elements 
arbitrarily in a co-ordinate screw ; the fifth will be determined by the 
condition that this screw shall be reciprocal to X When the body 
has five degrees of freedom it cannot be in equilibrium unless the 
forces which act upon it. form a wrench about X 
The cases of freedom //. and the reciprocal system of freedom IV 
are next discussed. If a body be capable of displacement about two 
screws A, B, it can be displaced about every screw upon the cyliudroid 
determined by ^, B. Thus the cylindroid expresses the freedom en- 
joyed by the body. A screw X in general cuts the cylindroid in three 
points, P, Q, R. If the screw X be perpendicular to the generator of 
the cylindroid through P, it will be found that the generators through 
Q, R have equal pitches. If now we assign to X a pitch equal in 
magnitude, but opposite in sign, to the pitches of the generators through 
Q and R, then X will be reciprocal to the cylindroid. Through any 
point 0 in space it is found that a whole cone of the second order of 
screws reciprocal to the cylindroid can be drawn. This cone is easily 
constructed, by joining the point 0 to the ellipse in which a certain 
tangent plane cuts the cylindroid. A body free to twist about the 
cylindroid will be in equilibrium, notwithstanding a wrench about any 
one of these reciprocal screws, a cone of which can be drawn through 
every point. The reciprocals of these theorems contain the most ge- 
neral solution of the problem of freedom IV. We learn that when a 
body is free to twist about four screws, a cylindroid can be uniquely 
determined reciprocal to these four screws ; when this cylindroid has 
been found, the entire co-ordinate system becomes known, and a cone 
of the second order of co-ordinate screws can be drawn through every 
point. We also see that the reactions of the system can only produce 
a wrench about a screw of the reciprocal cylindroid, or, in other words, 
that a body having four degrees of freedom can only be in equilibrium 
when the wrench which acts upon it is about a screw on the recipro- 
cal cylindroid. One screw co-ordinate with freedom IV. can be drawn 
through any point in space, so that its pitch shall have any given value 
from + 00 to - 00 , The case of freedom III is of peculiar interest, 
partly from the fact that the reciprocal system is of the same order, 
and also because the important problem of the motion of a body about 
a fixed point is included. In the present memoir, however, particular 
cases are not discussed. Three co-ordinate screws, Xi, Xo, X3, can be 
diawn through any point 0 in space co-ordinate with three given 
screws ^, ^, C. Three screws, Fj, Yo, F3, can also be drawn through 
0. reciprocal to A, B, C. If a sphere be described around 0 as centre, 
the spherical triangle formed by the intersection with the sphere of 
X,, X2, X3 is the polar of the triangle formed by the intersection of 
F,, F2, F3. A screw of given pitch p, reciprocal to the system A, B, C, 
must be a generator of a certain hyperboloid. The other system of 
generators of the hyperboloid is the locus of screws reciprocal to A, B, 
