Ball — Problems on the Dynamics of a Rigid Body. 553 
The kinetic energy of a body twisting about the screw a with the 
twist velocity Sa 
If a body receive an impulsive wrench about the screw 7i, whose 
coordinates are proportional to 
«iai, S2«2, 5303, «ia4, «5 05, S6«6» 
then the body commences to twist about an instantaneous screw a. 
The condition that the screws a and /3 be reciprocal is 
Si a, /3i + 52 a2 + ^3 «3 ^3 + h «4 + «5 % A + «6 ^6 A = 0 
The conditions that ?^ screws shall form a complex of a lower order 
can be expressed by the evanescence of a single function of the coordi- 
nates. 
The partial differential coefficients of the pitch of a screw with 
respect to the coordinates are proportional to the coordinates of the 
corresponding impulsive screw. 
If p be an impulsive screw, we find the corresponding instantaneous 
screw selected from the screw-complex (III. 1 ), as follows : 
Draw the reciprocal plane to and find the diameter of the ellip- 
soid of equal energy, which is conjugate to the reciprocal plane, then 
the screw of the complex parallel to this diameter is the screw re- 
quired. 
If a rigid body have five degrees of freedom, then the principal 
screws of inertia are determined as follows. 
Let 7 be the screw reciprocal to the freedom, then the coordinates 
of one of the principal screws of inertia are proportional to 
Vi Vi Vs Vi V5 V6 
Si-V S2- V S^- V S^-V Sq-V 
where v is one of the five roots of the equation, 
Si-V S^-V S^-V Si-V S5-V Se-V 
