236 ON THE GEOMETRICAL REPRESENTATION OF THE MOTION OF A SOLID BODY. [36 
whence u' = a p' + ¡3 q' + 7 r', 
v' = a' p' + (3' q' + 7' r', 
w' = a"p' + ¡3"q + 7V, 
(the remaining terms vanishing as is well known); and therefore 
vw' — v'w = a (qr' — q'r) + /3 (rp' — r'p) + 7 (pq' — p'q), 
wu' — w'u = a' (qr — q'r) + /3' (rp' — r'p) + 7' (p^' — p'g), 
uv' — u'v = a." (qr' — q'r) + /3" (rp' — r'p) + 7" (p*/ — p'p). 
Hence 
uw" — = a (qr" — q"r) + /3 (rp" — r"p) + 7 (pq" — p"q) + u'a> 2 — uaxo', 
wu" — w"u = a! (qr” — q'r) + /3' (rp" — r"p) + 7' (pg" — p"q) + v'ar — vaxo’, 
uv" — u"v = a." (qr" — q"r) + /3" (rp" — r"p) + 7" (pq" — p"q) + w'or — wcoco', 
and multiplying these by u', v', w', and adding, the required equation is immediately 
obtained. 
In fact, if r be the distance of a point in the instantaneous axis from the vertex, 
and p, a the radii of curvature of the two cones at that point, then 
r CO 3 „ r ft) 3 ^7 
p Jft <r 
as may be shown without difficulty: and the angular velocity of the instantaneous axis 
M1 
is given by the equation = —-; hence the relation between the two angular veloci 
ties is 
111 
ft) : ot = : — . 
par