22 
DR. R. S. BALL’S RESEARCHES IN THE DYNAMICS OF A RIGID 
(e) A body might receive six small twists about the six screws, so that after the last 
twist the body would occupy the same position which it had before the first. 
(/') By extending the language of Professor Sylvester (‘Comptes Bendus,’ t. lii. p. 741), 
we might perhaps assert that six screws are in involution when their sexiant vanishes. 
(g) Any wrench (or twist or twist velocity) can be resolved into six wrenches (or twists 
or twist velocities) about six screws when the sexiant of the six screws does not vanish. 
12. Application of the sexiant to the resolution of a wrench about six screws of 
reference . — One of the most remarkable properties of the sexiant is enunciated in the 
following theorem : — 
If seven wrenches (or twists) about seven screws equilibrate (or neutralize), the mag- 
nitude of each wrench (or twist) must be proportional to the sexiant of the remaining 
six screws. 
We shall demonstrate this property for twists, the demonstration for wrenches being, 
of course, exactly similar. 
Let one, A*, of the seven screws be represented by 
x—x k 
Ujc 
y—yk*—*k 
fik 7k 
(pitch =§*). 
Suppose x k , y k , z k be the coordinates of the foot of the perpendicular from the origin 
on A*. 
The body receives a small twist u k about A k ; the rotation element of the twist may 
be transferred to a rotation about a parallel axis through the origin by the introduction 
of translations, whose components are 
a k{7kyk — /3/ c %) > x k (a lc z k y k x k ), a k (j 3 k x k a k y k ). 
Each of the seven twists is thus decomposed into three translations parallel to the 
three axes, and three rotations about the axes. If the seven twists neutralize, we have 
the six equations : 
CO i<55 x -J- &C. “J- == 0 , 
+ &c. 0, 
a \7\ + & c - + u i7i — 0 5 
Zifr) + &c.+*> 7 (g r a ; +y 7 y 7 — ^ Z7&)=0, 
®i(§iPi J V z \ a >\ — x \7\) + &C. ~ 0, 
®i(g i7i ■ +& ifr — y ia x ) + &c. +^(§777 + x 7 (3 7 —y 7 u 7 ) =0. 
These equations will be satisfied if for each value of co the sexiant of the remaining 
six screws be substituted. This will appear most satisfactorily by writing one of the 
equations a second time and eliminating co L See., u 7 from the seven equations. We then 
get a result which is necessarily zero. But it will be found that this is precisely the 
same result as would have been obtained by the substitution already mentioned. 
13. Circumstances under which the sexiant vanishes identically . — The sexiant vanishes 
if k- 1-1 screws be members of a coordinate system of freedom k [art. 31]; for then a 
