Page images

Composition ofangu. lar veloci: ties about Axes meeting in a point.

Now the perpendicular from any point x, y, z to this line is, by co-ordinate geometry,

(x+py + oz)*73

Do + på +0
Jos + p +0

1 (oz-oy)+ (ox – az)+ (@y px)'
whole displacement of x, y, z

Nas + p + o? dt The actual displacement of x, y, z is therefore the same as would have been produced in time st by a single angular velocity, N=VW+ + p + o*, about the axis determined by the preceding equations.

Composi. tion of suc

97. We give next a few useful theorems relating to the cessive finite composition of successive finite rotations. rotations.

If a pyramid or cone of any form roll on a heterochirally similar* pyramid (the image in a plane mirror of the first position of the first) all round, it clearly comes back to its primitive position. This (as all rolling of cones) is conveniently exhibited by taking the intersection of each with a spherical surface. Thus we see that if a spherical polygon turns about its angular points in succession, always keeping on the spherical surface, and if the angle through which it turns about each point is twice the supplement of the angle of the polygon, or, which will come to the same thing, if it be in the other direction, but equal to twice the angle itself of the polygon, it will be brought to its original position.

The polar theorem (compare $ 134, below) to this is, that a body, after successive rotations, represented by the doubles of the successive sides of a spherical polygon taken in order, is restored to its original position; which also is self-evident.

98. Another theorem is the following:

If a pyramid rolls over all its sides on a plane, it leaves its track behind it as one plane angle, equal to the sum of the plane angles at its vertex.

* The similarity of a right-hand and a left-hand is called heterochiral: that of two right-hands, homochiral. Any object and its image in a plane mirror are heterochirally similar (Thomson, Proc. R. S. Edinburgh, 1873).



about a fixed

ing cones,

Otherwise :-in a spherical surface, a spherical polygon having Composition rolled over all its sides along a great circle, is found in the sive finite same position as if the side first lying along that circle had been simply shifted along it through an arc equal to the polygon's periphery. The polar theorem is:-if a body be made to take successive rotations, represented by the sides of a spherical polygon taken in order, it will finally be as if it had revolved about the axis through the first angular point of the polygon through an angle equal to the spherical excess ($ 134) or area of the polygon.

99. The investigation of $ 90 also applies to this case; and it Motion is thus easy to show that the most general motion of a spherical point. Rollfigure on a fixed spherical surface is obtained by the rolling of a curve fixed in the figure on a curve fixed on the sphere. Hence as at each instant the line joining C and O contains a set of points of the body which are momentarily at rest, the most general motion of a rigid body of which one point is fixed consists in the rolling of a cone fixed in the body upon a cone fixed in space—the vertices of both being at the fixed point.

100. Given at each instant the angular velocities of the Position of body about three rectangular axes attached to it, determine to given roits position in space at any time.

From the given angular velocities about 0A, OB, OC, we know, $ 95, the position of the instantaneous axis OI with reference to the body at every instant. Hence we know the conical surface in the body which rolls on the cone fixed in space. The data are sufficient also for the determination of this other cone; and these cones being known, and the lines of them which are in contact at any given instant being determined, the position of the moving body is completely determined.

If , M, v be the direction cosines of OI referred to 0A, OB,
OC ; a, p, o the angular velocities, and w their resultant:


by $ 95. These equations, in which w, p, o, w are given functions
of t, express explicitly the position of OI relatively to 01, OB,

р M


[ocr errors]

Position of the body due to given rotations,

OC, and therefore determine the cone fixed in the body. For
the cone fixed in space : if r be the radius of curvature of its
intersection with the unit sphere, go? the same for the rolling
cone, we find from $ 105 below, that if s be the length of the
arc of either spherical curve from a common initial point,
1 ds

1 ds


r – + go dt

pdt which, as 8, r' and w are known in terms of t, gives q in terms of t, or of 8, as we please. Hence, by a single quadrature, the

“intrinsic” equation of the fixed cone. 101. An unsymmetrical system of angular co-ordinates 4,0,0, for specifying the position of a rigid body by aid of a line OB and a plane A OB moving with it, and a line OY and a plane YOX fixed in space, which is essentially proper for many physical problems, such as the Precession of the Equinoxes and the spinning of a top, the motion of a gyroscope and its gimbals, the motion of a compass-card and of its bowl and gimbals, is convenient for many others, and has been used by the greatest mathematicians often even when symmetrical methods would have been more convenient, must now be described.

ON being the intersection of the two planes, let YON=%, and NOB =$; and let O be the angle from the fixed plane, produced through ON, to the portion NOB of the moveable plane. (Example, the "obliquity of the ecliptic,” f the longitude of the autumnal equinox reckoned from OY, fixed line in the plane of the earth's orbit supposed fixed; & the hour-angle of the autumnal equinox; B being in the earth's equator and in the meridian of Greenwich : thus 4, 0, $ are angular co-ordinates of the eartb.) To show the relation of this to the symmetrical system, let OA be perpendicular to OB, and draw OC perpendicular to both; 0X perpendicular to OY, and draw OZ perpendicular to OY and OX; so that OA, OB, OC are three rectangular axes fixed relatively to the body, and OX, OY, OZ fixed in space. The annexed diagram shows 4, 0, $ in angles and arc, and in arcs and angles, on a spherical surface of unit radius with centre at 0.

To illustrate the meaning of these angular co-ordinates, suppose A, B, C initially to coincide with X, Y, Z respectively.


Then, to bring the body into the position specified by 0, 0, 4, Position of rotate it round OZ through an angle equal to f+$, thus due to given

[merged small][merged small][ocr errors][ocr errors][merged small][ocr errors][merged small]

bringing A and B from X and Y to A' and B' respectively;
and, (taking YŇ=%,) rotate the body round on through an
angle equal to 0, thus bringing A, B, and C from the positions
A', B, and Z respectively, to the positions marked A, B, C in
the diagram. Or rotate first round ON through 0, so bringing
C from Z to the position marked C, and then rotate round
OC through 4 +0. Or, while OC is turning from OZ to the
position shown on the diagram, let the body turn round OC
relatively to the plane ZcZ'O through an angle equal to $.
It will be in the position specified by these three angles.

Let 2 XZC=\, .ZCA = T - $, and ZC = 0, and w, p, o mean
the same as in g 100. By considering in succession instantaneous
motions of Calong and perpendicular to ZC, and the motion of
AB in its own plane, we have

=w sin $ + p cos $, sin A =p sin $-w cos ,

dt dy


cos O +

The nine direction cosines (XA), (YB), &c., according to the
notation of $ 95, are given at once by the spherical triangles


= 0. dt

Position of the body due to given rotations.

XNA, YNB, &c.; each having N for one angular point, with 0,
or its supplement or its complement, for the angle at this point.
Thus, by the solution in each case for the cosine of one side in
terms of the cosine of the opposite angle, and the cosines and
sines of the two other sides, we find

(XA)= cos 6 cos y cos 6-siny sin ,
(XB) = cos O cos y sin p-sin y cos ,
(YA) = cos 0 sin y cos $ + cos y sin $.

(YB) cos 6 sin y sin $ + cos y cos ,
(YC) =

sin sinų,
(ZB) = sin 6 sin .

cos ,

General motion of a

(ZC) =
(ZA) = - sin 0 cos ,

(XC) = sin 0 cosy. 102. We shall next consider the most general possible motion rigid body. of a rigid body of which no point is fixed—and first we must

prove the following theorem. There is one set of parallel planes in a rigid body which are parallel to each other in any two positions of the body. The parallel lines of the body perpendicular to these planes are of course parallel to each other in the two positions.

Let C and C be any point of the body in its first and second positions. Move the body without rotation from its second position to a third in which the point at C” in the second position shall occupy its original position C. The preceding demonstration shows that there is a line CO common to the body in its first and third positions. Hence a line C'O' of the body in its second position is parallel to the same line CO in the first position. This of course clearly applies to every line of the body parallel to CO, and the planes perpendicular to these lines also remain parallel.

Let S denote a plane of the body, the two positions of which are parallel. Move the body from its first position, without rotation, in a direction perpendicular to S, till S comes into the plane of its second position. Then to get the body into its actual position, such a motion as is treated in 8 79 is farther

« PreviousContinue »