Ball — Kinematics and Dynamics of a Rigid System. 255 



correspond witli vertices so related together, a^ is indeed one of the 

 screws about which the system is translated. 



Let 6 be the distance along the ray a/3 by which a point is dis- 

 placed, then if the coordinates of that point be 



ttl + AySl, &c., 



we have 



n = 2A(a, /3i + ao /?o + aa ^3 + aj, ^4) : 



making the same substitution in Z7it reduces to 



A f p + - J (tti /3i + tto ^3 + tta /?3 + ai/Ji), 



whence we obtain finally 



cos^ = |f p + - j, 



whence we deduce iO = log p, 



so that the pitch of the displacement is 



log Pi 

 log pi 



where pj and p, are two not reciprocal roots of the equation 



p* + 4^p3 + 6^p2 + 4:Ap +1=0. 



We may here notice a proof that the surface ZT" passes through the 

 four generators belonging to the tetrahedron. Let a and /? be the ver- 

 tices corresponding to p and p', then by substitution in Cwe find 



■^ (P + p') (<^i Pi + «2 Pt + 0.3/33 + aifSi), 



and in O 2X{ai(3i + ao/Jn + 0.3/3-3 + ai/3i), 



but if a;8 be a generator of CI, then the last line must vanish, and so 

 must the former one also. 



If the movement be a vector, then the equation for p reduces to the 

 form 



(p^ - 2p cos a + 1)". 



It is well known that when twists about two given screws are 

 compounded together they will constitute a twist on a single screw, 

 which is a generator of a ruled surface of the fourth order (Linde- 

 mann). The laws of the distribution of pitch on the generators of this 

 surface are now to be given. 



