18 
DE. E. S. BALL’S EESEAECHES IN THE DYNAMICS OE A EIGrlD 
It follows, from art. (3), that precisely the same investigation determines seven twist 
velocities about seven screws which neutralize each other. 
II. COEECIPEOCAL SCEEWS. 
5. On a property of five twists analogous to a property of two vectors . — That one 
vector y can be determined, which is perpendicular to two given vectors a, (3, is a 
proposition to which an analogue may he found in the Theory of Screws. The 
mechanical equivalent of the simple vector theorem just referred to expresses that a 
force directed along the vector y is unable to disturb the equilibrium of a particle only 
free to be displaced parallel to a and 0, or, in fact, to move in the plane of a and (3. 
We may state the result thus: a particle which has only one degree short of absolute 
freedom can only be in equilibrium when the force acting on the particle occupies one 
position. 
To the theorem expressed in this manner, we have an analogous proposition in the 
Theory of Screws : a rigid body which has only one degree short of absolute freedom 
can only be in equilibrium when the wrench acting on the body is about one determi- 
nate screw. This is demonstrated as follows. The body having only one degree short 
of absolute freedom must he capable of twisting about five screws [art. 94]. Any 
wrench which is unable to disturb the equilibrium of the body must be reciprocal to 
the five screws ; and since a screw is determined by five elements, only a finite number 
of wrenches fulfilling this condition are possible. Now, as pointed out [art. 45], that 
number must he one ; for if there were two, then every screw on the cylindroid deter- 
mined by those two would fulfil the conditions [art. 21], and the number would be 
infinite. 
6. Calculation of the single screw reciprocal to five given screws. — Let one of the five 
given screws be typified by 
x—Xk_y—yj 1 
*k ~ file 
while the desired screw is defined by 
x — x ] y — y 1 
(pitch =§*), 
„ _r 
— (P itch =§)• 
The condition of reciprocity (art. 1) produces five equations of the following type : — 
«[(§ + §*)«*+ yi&ic— &»*] + (3[(§ + e*)0* + 
+ y[(g + u&J 4- «k(yy' — fid) +£*( az' — yrf) 
+ y k (l3%'—ay')=0. 
From these five equations the relative values of the six quantities 
a, 13, 7, yy'—(3z', az' — yx', fix’ —ay' 
can be determined by linear solution. Introducing these values into the identity 
*(yy'-PzW(xz'-yx , )+y(l3x'-*y')= 0 , 
gives the equation which determines g. 
