Ball— Small Oscillations of a Rigid Body. 235 
be regarded as a degraded form of the cylindroid {m = o) from which 
the most essential feature (the distribution of pitch) has disappeared. 
If a body only free to twist about a screw A be held in equilibrium 
by a wrench about the screw 2i, then conversely a body only free to 
twist about the screw B will be held in equilibrium by a wrench about 
A. This is called the theorem of the reciprocal screws. 
A pair of screws in space whose pitches are p, p' are inclined at an 
angle ^, and the length of the common perpendicular is d. The con- 
dition that these screws be reciprocal is 
{p ^- p') cos 0 - d sin 6= 0 
J^.B. — 0 is the angle between portions of the screws that follow the 
direction of a right-handed screw along the common perpendicular. 
Given a screw 8 and a point 0, through 0 draw OP, equal to the 
sum of the pitches of the two reciprocal screws along 8 and OP, the 
locus of P is an unclosed surface of the fourth order, called the 
pe ctenoid. 
The equation of the pectenoid is 
All pectenoids are similar surfaces, depending upon a single parameter. 
The screw S {y= o x = + a) called the cardinal screw of the pecte- 
noid. A perpendicular from the origin on is a nodal line of the 
surface. 
All plane sections of the pectenoid through the axis of x are circles ; 
hence all the screws of given pitch which can be drawn through a 
given point 0 reciprocal to a given screw lie in a plane. 
To analyse the freedom enjoyed by a rigid body we inquire about 
what screws in space the body can be twisted. 
If the body can be twisted about K screws, it can be twisted about 
an infinite number of screws which are called the co- ordinate system of 
freedom K. 
All the screws reciprocal to a co-ordinate system of freedom ^form 
a co-ordinate system of freedom 6 - iT. 
K{Q-K) data are necessary to define a co-ordinate system of 
freedom K. 
If, after all the screws in space have been tried, the body cannot be 
twisted about any screw, the body must be fixed in space. If the body 
can be twisted about only one screw S, the body has freedom /. 
Any line P in space may be a reciprocal screw to S, provided P have 
the proper pitch. It follows, by the principles of reciprocal screws^ 
that when the -fi^freedom equilibrium problem has been solved, the jDrob- 
lemof 6 - ^freedom is determined also. We therefore learn that when 
a body has five degrees of freedom the body can be twisted about any 
screw in space, provided that screw has the proper pitch. That pitch; 
is to be determined as follows : — Suppose A^, &c., A^, be the five screws 
