BODY BY THE AID OE THE THEORY OF SCREWS. 
27 
Combining the two sets of equations, we have for the specific impulsive forces 
F, See., F i+1 , 
*i±2. 
fil m — rn 
1 1 ±2 J -fc+l 
Sj Sg 
23. Determination of the specific impulsive wrenches when the three impulsive screws 
lie on a cylindroid, and the corresponding instantaneous screws are known. — In the case 
where k= 2 the problem of the last article assumes a simple form. 
Let $i, 0 2 , 0 3 be the angles corresponding to the instantaneous screws on the instantaneous 
cylindroid. Let <p 15 <p 2 , <p 3 be the angles corresponding to the impulsive screws on the 
impulsive cylindroid. 
If the following equations be true, 
C0 2 C0 3 
sill (0 2 — y sin (6 S — y sin (6 J — 0 2 ) ’ 
then we must also have 
F1CO1 ^ f> 2 __ F 3 cq 3 _ 
sin ( < f S 2 — ^ 3 ) sin (p 3 — Pi) sin (Pi-<P 2 ) ’ 
whence the relative values of F,, F 2 , F 3 are known. 
24. Delations between the impulsive screw and the instantaneous screw in certain special 
cases. — When an impulsive force acts upon a free quiescent rigid body, the directions of 
the force and of the instantaneous screw are parallel to a pair of conjugate diameters in 
the momental ellipsoid. 
When an impulsive wrench acting on a free rigid body produces an instantaneous 
rotation, the axis of the rotation must be perpendicular to the impulsive screw. 
When an impulsive force acting on a free rigid body produces an instantaneous rota- 
tion, the direction of the force and the axis of the rotation are parallel to the principal 
axes of a section of the momental ellipsoid. 
Y. THE PRINCIPAL SCREWS OF INERTIA. 
25. On the locus of the impulsive screw corresponding to a given instantaneous screw. 
— If an impulsive screw be given, the corresponding instantaneous screw is determinate, 
whether the body be free or constrained. If, however, the instantaneous screw be given, 
the impulsive screw is indeterminate, except in the case where the body is free. We 
have therefore the following problem : — To determine the locus (in a generalized sense) 
of an impulsive wrench which will cause a quiescent rigid body to commence to twist 
about a given screw. 
Let A 1 &c., A* be a group of screws defining the freedom of the body. 
Let Bj &c., B 6 _j be screws reciprocal to the freedom. 
S is the screw about which the body is to twist. 
Let P be any impulsive wrench which would make the body commence to twist about 
S. Then any screw X coordinate with the 7— k screws, 
P, Bj, &c., B 6 _jc, 
possesses the property required. This condition expresses the “ locus ” of X. 
e 2 
