26 
DE. E. S. BALL’S EESEAECHES IN THE DYNAMICS OE A EIGID 
20. On impulsive and instantaneous coordinate systems. — Some of the preceding results 
may be generalized. If impulsive wrenches were to act upon a partially or entirely free 
rigid body about the different screws belonging to a coordinate system of freedom k, 
then the corresponding instantaneous screws, and also the wrenches produced by 
the initial reactions of the constraints, would each constitute a coordinate group of 
freedom k. 
21. On a special impulsive cylindroid. — If the impulsive cylindroid be defined by a 
pair of screws along the same straight line, the surface becomes evanescent, and every 
screw on the straight line is contained on the cylindroid. We thus deduce the following 
theorem as a particular case of art. (17) : — 
If a rigid body, free or constrained in any way, were to receive impulsive wrenches 
about screws of every pitch on a straight line, then all the corresponding instantaneous 
screws lie on a cylindroid. 
One of the screws of zero pitch on the instantaneous cylindroid, when the body is 
free, passes through the centre of inertia of the body, because this must be the instan- 
taneous screw corresponding to infinite pitch along the impulsive screw. 
22. Calculation of the specific impulsive wrenches when a sufficient number of pairs 
of corresponding impulsive and instantaneous screws are known. — By the words specific 
impulsive wrench we are to understand an impulsive wrench which, acting on a free 
quiescent rigid body, will communicate to the body the unit of twist velocity about an 
instantaneous screw. 
Let A! &c., A* +1 be &+1 screws of a coordinate system of freedom k , and &> x See., u k+1 
be a set of twist velocities about A x See., A* +1 which neutralize. Let X x 8ec., X 4+1 be 
corresponding impulsive screws for a free body ; it is required to find the specific impul- 
sive wrenches. 
Let the impulsive wrenches be 
F 1 a 1 &c., F* + x u k+1 . 
Bj &c., B 6 _ 4 are screws reciprocal to the system A! Sec., A*. 
Q x &c., Q 6 _* are screws reciprocal to the system X, &c., X*. 
Adopting the notation for sexiants employed in art. (14), 
S,=S(A 2 &c., A* +i , Bj &c., Re.*), 
Tj=S(X 2 &c., X* +1 , Qi See., Q 6 _*), 
we have the equations 
Si 
^2 o “'fc + l 
— q — — OvC. — (5 * 
Oo 
Since the body is free, the wrenches must equilibrate which could produce these 
velocities that neutralize ; whence 
i 
T, 
=&c.= : 
k+ lWft+l 
17+ 1 
