BODY BY THE AID GE THE THEOBY OE SCEEWS. 
29 
property that an impulsive wrench about each of them will make the body commence 
to twist about the same screw. 
We shall state the argument for three degrees of freedom, the general argument for 
any other degree of freedom being precisely similar. 
Let A n A 2 , A 3 be the three screws of the system which we have just determined. 
Bj, B 2 , B 3 are any corresponding impulsive screws. 
R 1? R 2 , R 3 are any three screws reciprocal to A 1; A 2 , A 3 [art. 37]. 
Any impulsive wrench about a screw coordinate with B 1? R„ R 2 , R 3 will make the 
body twist about A x (art. 25). But the screws of the coordinate system containing 
B„ R 1? R 2 , R 3 are defined by being reciprocal to A 2 and A 3 . Now A 1 is reciprocal to 
A 2 , A 3 , and therefore Aj must be coordinate with B 1? R n R 2 , R 3 [art. 89]. Therefore an 
impulsive wrench about A l will make the body twist about A,. 
It can be shown, by similar reasoning to that employed in art. (26), thaf the It principal 
screws of inertia are unique. 
29. On the pitch hyperbola in the principal plane of a cylindroid . — We shall exemplify 
the preceding articles by determining the principal screws of a rigid body which has two 
degrees of freedom. The cylindroid expressing the freedom being drawn, we shall give 
the construction by which the two screws on that cylindroid may be found which possess 
the property in question. 
To obtain, in the first place, a clear view of the distribution of the pitch upon a cylin- 
droid, we introduce a conic which is to be known as the pitch hyperbola. 
The cylindroid having as its equation 
z{x 2 -\-y 2 ) — 2mxy = 0 [art. 7], 
the generator in which the plane 
y=x tan 6 
cuts the surface has a pitch 
cos 23. 
The pitch hyperbola has for its equation 
X 2 
m—p m-\-p 
If q be the radius vector of the pitch hyperbola in the direction 3, 
p-\-m cos 2Q= --( m ^ ; 
hence we have the fundamental property of this conic, which is thus enunciated : — 
The pitch of any generator of the cylindroid is proportional to the inverse square of 
the parallel diameter in the pitch hyperbola. 
Given a screw A on a cylindroid, another screw B on the cylindroid reciprocal to A 
can always be determined [art. 44]. We shall now show that 
A pair of reciprocal screws on a cylindroid are parallel to a pair of conjugate diameters 
in the pitch hyperbola. 
