BODY BY THE AID OE THE THEORY OE SCREWS. 
31 
By the properties of what we have called conjugate screws of kinetic energy [art. 56], 
we now see that every pair of conjugate diameters of the ellipse of equal kinetic energy 
are parallel to a pair of conjugate screws of kinetic energy on the cylindroid. 
31. Construction of the principal screws of inertia for a rigid body with two degrees 
of freedom. — Draw the pitch hyperbola and the ellipse of equal kinetic energy; since 
these conics are concentric, a pair of common conjugate diameters can be drawn. The 
screws upon the cylindroid parallel to the common conjugate diameters are the principal 
screws of kinetic energy. 
Let A„ A 2 be the two screws thus determined, and let X 1? X 2 be the impulsive screws, 
wrenches about which, if the body were free, would make it commence to twist about 
A„ A 2 . Let B,„ R 2 , B, 3 , R 4 be any four screws reciprocal to the cylindroid. 
An impulsive wrench about any screw coordinate with X 1? 11,, B 2 , B 3 , R 4 will make 
the body twist about A,. But since the screws are conjugate screws of kinetic energy, 
X, is reciprocal to A 2 . Thus A 2 is reciprocal to the five screws X„ It,, R 2 , R 3 , R 4 ; and 
every other screw reciprocal to A 2 will therefore be coordinate with the five screws just 
written. 
Since A, and A 2 are parallel to conjugate diameters of the pitch hyperbola, A, is reci- 
procal to A 2 , and therefore coordinate with the five screws ; an impulsive wrench about 
A, will therefore make the body commence to twist about A,. In a similar manner it 
can be shown that A 2 is the other principal screw of inertia. 
32. Relation between a twist about a screw on a cylindroid and its components along 
a pair of reciprocal screws. — Let a be the twist, and &> 1} u. 2 the components. The pitch 
hyperbola referred to axes parallel to the screws corresponding to u 2 has for equation 
xf yf 
a n^T b n- 1 - 
If § be the radius vector parallel to the screw appropriate to a, 
CO CO, C0 2 . 
y’ 
whence 
l 2 . 1 2 1 2 
If ftirRurR be the pitches of &>„ a> 2 , &>, we have 
pA+p&l=pu\ 
This result may be compared with art. (10). 
We easily foresee, what a little calculation will verify, that, if the virtual coefficient 
of cy„ u 2 , instead of vanishing, had the value B, the relation just written must be 
replaced by 
p x u\ -j- Rcy,cy, -{-p 2 al=pco 2 . 
Here, again, we are reminded of the analogy between the cosine and the virtual coeffi- 
cient. This result may be generalized into a theorem which is of considerable interest. 
Let A, &c,, Aj. be any Ic screws of pitches p^ &c., p k . 
