There are therefore three permanent screw-motions such that the corresponding impulsive wrench in each case reduces to a couple only. The axes of these three screws are mutually at right angles, but do not in general intersect.

115. We will now shew that in all cases where the impulse consists of a couple only, the motion can be completely determined. It is convenient, retaining the same directions of the axes as before, to change the position of the origin. To transfer the origin to any point (x, y, z) we must write

for u, v, w, respectively. We have then in the expression for the kinetic energy

new M" - Bx + M", new N' Cx + N', &c.,

Let us denote the

=

N L"

M

N'

=

=

C'

C A

A

B'

values of these pairs of equal quantities by

a, B, y respectively. The formulæ (26) may then be written

=

L
- p2 +
A

M'

B

N"

(32),

q2+ -r2 + 2xqr + 2ßrp +2ypq..............(33). C

The motion of the body at any instant may be conceived as made up of two parts;-a motion of translation equal to that of the origin, and one of rotation about an instantaneous axis passing through the origin. The latter part is to be determined by the equations

These are identical in form with the equations of motion of a rigid body about a fixed point, so that we may make use of Poinsot's well-known solution of the latter problem. The angular motion of the body is therefore obtained by making the ellipsoid (30), which is fixed in the body, roll on the plane

x + μy + vz = const.,

which is fixed in space, with an angular velocity proportional to the length OI of the radius vector drawn from the origin to the point of contact I. The representation of the actual motion is then completed by impressing on the whole system of rolling ellipsoid and plane a velocity whose components are given by (32). The direction of this velocity is that of the normal OM to the tangent plane to the quadric

Ψ (x, y, z) = - .

.....(34),

at the point P where Of meets this quadric, and its magnitude is × angular velocity of body.........(35).

OP.OM If OI do not meet (34), but the conjugate quadric obtained by changing the sign of e, the sense of the velocity (35) is reversed.

116. Of course for particular varieties of the moving solid the expression for 27' becomes greatly simplified. For instance:

(a) let us suppose that the body has a plane of symmetry as regards both its form and the distribution of matter in its interior, and let this plane be taken as that of xy. It is plain that the energy of the motion is unaltered if we reverse the signs of w, p, q, the motion being exactly similar in the two cases. This requires that A', B', P', Q', L, M, L', M', N" should vanish. One of the directions of permanent translation is then parallel to z. The three screws of Art. 114 are now pure rotations; the axis of one of them is parallel to z; those of the other two are at right angles in the plane xy, but do not in general intersect the first.

(b) If the body have a second plane of symmetry, at right angles to the former one, let this be taken as the plane of zx. We find in the same way that in this case the coefficients C', R', N, L" also must vanish, so that the expression for 2T

assumes the form

2T = Au2 + Bv2 + Cw2

+ Pp2 + Qq2 + Rr2

+ 2N ́wq +2Μ"vr.

............(36).

The directions of permanent translation are parallel to the three axes of co-ordinates. The axis of x is the axis of one of the permanent screws (now pure rotations) of Art. 114; and those of the other two intersect it at right angles (being parallel to y and z respectively), though not necessarily in the same point.

(c) If, further, the body be one of revolution, about æ, say, the value of 2T given by (35) must be unaltered when we write v, q, - w, -r for w, r, v, q, respectively; for this is merely equivalent to turning the axes of y, z through a right angle. Hence we must have B = C, Q = R, M" = − N'. If we further transfer the origin to the point O of Art. 115 we have M" - N'. These conditions can be satisfied only by M" = 0, N' = 0, so that

2T = Au2 + B(v2 + w2)

+ Pp2 + Q(q2+r2)...........

.(37).

(d) If in (b) the body have a third plane of symmetry at right angles to the two former ones, then taking this plane as that of yz we have, evidently,

2T = Au2 + Bv2 + Cw2

+ Pp2 + Qq2 + Rr2........(38).

The axes of co-ordinates are in the directions of the three permanent translations; they are also the axes of the three permanent screwmotions (now pure rotations) of Art. 114.

(e) Next let us consider another class of cases. Let us suppose that the body has a sort of skew symmetry about a certain axis (say that of x), viz. that it is identical with itself turned through two right angles about this axis, but has no plane of symmetry*. The expression for 27 must be unaltered when we

change the signs of v, w, q, r, so that the coefficients B', C', Q', R', M, N, L', L" must all vanish. We have then

2T=Au2 + Bv2 + Cw2 + 2A'vw

+Pp2 + Qq2 + Rr2 + 2P'qr

+2Lpu

+2q (M'v + N'w)

+2r (M”v + N'w)..

.....

(39).

The axis of x is one of the directions of permanent translation; and also the axis of one of the three screws of Art. 114, the pitch

being

L
A

The axes of the two remaining screws intersect it

at right angles, but not in general in the same point.

(f) If, further, the body be identical with itself turned through one right angle about the above axis*, the expression (39) must be unaltered when v, q, — w, r are written for w, r, v, q, respectively. This requires that B = C, A′ = 0, Q = R, P′ = 0, M'=N", N'=-M". If we further transfer the origin to the point chosen in Art. 115 we must have N'=M", and therefore N' = 0, M" = 0. Hence (39) becomes

2T = Au2 + B(v2 +w3)

+Pp3 + Q(q2+r2)
+ 2 Lpu

+2M' (vq + wr).....

.(40).

(g) If the body possess the same properties of skew symmetry about an axis intersecting the former one at right angles, we evidently must have

2T = A (u3 + v3 +w2)

+P(p2 + q2 + r2)

+2L (pu + qv+rw)..

.(41).

Any direction is now one of permanent translation, and any line drawn through the origin is the axis of a screw of the kind con

Ꮮ

sidered in Art. 114, of pitch -4.

A'

The form of (41) is unaltered

Α

117. In the case (c) of a solid of revolution, the complete determination of the motion (when no external forces act) has been shewn by Kirchhoff* to be reducible to a matter of quadratures.

The particular case where the solid moves without rotation about its axis of symmetry, and with this axis always in one plane (i.e. when p = 0, q = 0), has been examined at length by Thomsont and Kirchhoff. The equations (24) then become

Let X, Y be the co-ordinates at any instant of the moving origin relatively to axes fixed in space in the plane xy, the direction of X being that of the resultant impulse I of the motion; and let denote the angle (measured in the positive direction) which makes with X. We have then

Au = I cos 0, Βυ = - I sin 0, r = 0,

The first two of equations (42), which merely express the fixity of the direction of the impulse in space, are satisfied identically; the third gives

the equation of motion of a common pendulum. When has been determined so as to satisfy (43) and the initial conditions, X and Y

are to be found from the equations

X = 4 cos - v sin 0 = 11(1+1)+(一)

,

u

+B

cos 9,

...(44),