1888.] in a Inquid when the impulse reduces to a couple. 271 
and the axes of steady motion are parallel to those of the quadric 
T” (x, y, 2) = const. 
We may now show how by a choice of origin u, v, w can be- 
come the partial differential coefficients with respect to p, g, 7 
of a quadratic function of p,q,r. If we take any point (a, y, 2) 
as origin, then in 27’ we must write 
Ut Ty —Q2Z, V+ pZ—TL, Wt Que— py 
instead of wu, v, w, and the coefficients in the new energy function 
will be found by writing 
Cog — Cop@, Cys oF Ca5%, for Cyogr Co5 
and similar expressions with y, and z for the other coefficients 
of the same sort. If then 
¢, C C 
Qn =< — C8 Dy = Oe Le 
Cop 33 33 il Cy Coo 
/ ! 
Ge hE , sats 
we shall have 2& = * =a say, where ¢,,’ is the new c,,, and similar 
26? 
22 33 3 : 
expressions for the other new coefficients. In this case 
a GB's DONE et ge 
ges apace ec 
— Cis Cos Cog 7 : 
where 20 = ( eee a, B, vp, q r) ° 
The point here chosen as origin is the centre of the parallel- 
epiped whose three alternate edges are the possible axes of steady 
motion with no force-resultant of impulse. 
Lamb’s construction for the motion is then as follows :—Con- 
struct an ellipsoid 7” (x, y, z)=const. whose centre is at the 
point above determined, and let it roll on a plane which 
is parallel to the plane of the resultant impulsive couple (in 
this case the whole impulse of the motion), with an angular 
velocity proportional to the radius vector of the point of con- 
tact ; if the ellipsoid start with its principal axes parallel to the 
axes of the possible steady motions, they will always remain 
so. Next draw the quadric O(a, y, z2)=const., and let the in- 
stantaneous axis OJ about which the solid is rotating, cut this 
quadric in P, and draw the central perpendicular OM on the 
tangent plane at P; then the whole system is to be moved 
forward with linear velocity which varies mversely as the product 
OP . OM, and whose direction at any instant is parallel to MO. 
20—2 
