is very complicated. That axis of rotation, of which all the points remain at rest during the time dt, is called an instantaneous axis of rotation, for the solid revolves about it during that short interval, as it would do about a fixed axis.

The equations (32) must now be so transformed as to give all the circumstances of rotatory motion.

194. The equations in article 163, for changing the co-ordinates, will become

x = ax

+ by

+ cz'

where a, b, c are the cosines of the angles made by x with a', y', z'; a', b', c' are the cosines of the angles made by y with r', y', z'; and a", b", c" are the cosines of the angles made by z with the same axes Whatever the co-ordinates of dm may be, since they have the same

origin,

x2+ y2 +

z = x2 + y + z12.

By means of these it may be found that

a2 + a + a/2 = 1

b2 + b2 + b2 = 1

c + c + c = 1

ab + a'b' + a'b" = 0

ac + a'c' + a"c" = 0

bc + b'c' + b'c' = 0.

In the same manner, to obtain æ', y', z', in functions of x, y, z,

six of the quantities a, b, c, a', b', c', a', b'', c", are determined by

the preceding equations, and three remain arbitrary.

If values of æ', y', z', found from equations (36) be compared with

their values in equations (37), there will result

195. The axes x', y', z' retain the same position in the interior of the body during its rotation, and are therefore independent of the time; but the angles a, b, c, a', b', c', a", b", c", vary with the time; hence, if

If a', a", b', &c. be eliminated from this equation by their values in

by the same process it may be found that

a'Ap + b'Bq + c'Cr = SM'dt,

a"Ap + b"Bq + c"Cr=SM'dt.

196. If the differentials of these three equations be taken, making all the quantities vary except A, B, and C, then the sum of the first differential multiplied by a, plus the second multiplied by a', plus the third multiplied by a", will be

in consequence of the preceding relations between a a'a', bb'b", cc' c'', and their differentials. By a similar process the coefficients bbb", &c., may be made to vanish, and then if

aM + a'M' + a" M" = N

And if a, a', a', b, b', &c., and their differentials, be replaced by their functions in 4, 4, and o, given in article 194, the equations (39)

become

cos 0.dy.

197. These six equations contain the whole theory of the rotation of the planets and their satellites, and as they have been determined in the hypothesis of the rotatory pressures being zero, they will give their rotation nearly about their principal axes.

198. The quantities p, q, r determine oz", the position of the real and instantaneous axis of rotation, with regard to its principal axis oz'; when a body has no motion but that of rotation, all the points in a permanent axis of rotation remain at rest; but in an instantaneous axis of rotation the axis can only be regarded as at rest from one instant to another.

If the equations (36) for changing the co-ordinates, be resumed, then with regard to the axis of rotation,

dx = 0, dy = 0, dz = 0,

since all its points are at rest; therefore the indefinitely small spaces moved over by that axis in the direction of these co-ordinates being zero, the equations in question become,

:

x'da + y'db + z'dc = 0,

x'da' + y'db' + z'de' = 0,

x'da" + y'db" + z'dc" = 0,

which will determine x, y, z', and consequently oz" the axis in question.

For if the first of these equations be multiplied by c, the second by c', and the third by c", their sum is

Again, if the first be multiplied by b, the second by b', and the third by b", their sum is

Lastly, if the first equation be multiplied by a, the second by a', and the third by a", their sum is

qz' - ry' = 0.

The last of these is contained in the two first, which are the equations to a straight line oz", which forms, with the principal axes a',

then if x', y', z', be the co-ordinates of the point z",

;

Consequently oz" is the instantaneous axis of rotation.

fig. 51.

201. The angular velocity of rota

tion is also given by these quantities.

for the components of the velocity of a particle; hence the resulting

velocity is

√do + d sin 20

√ q2 +

2

dt

which is the sum of the squares of the two last of equations (41). 199. But in order to obtain the angular velocity of the body, this quantity must be divided by the distance of the particle at c' from the axis oz"; but this distance is evidently equal to the sine of z'oc, the angle between oz' and oz", the principal and instantaneous axes of rotation; but

and therefore

1

or

+++", , is the sine;

p2 +

√I2 + p
√p2 + q2 + r2

2

P

2

p2 + q + r2 is the angular velocity of rotation. Thus, whatever may be the rotation of a body about a point that is fixed, or one considered to be fixed, the motion can only be rotation about an axis that is fixed during an instant, but may vary from one instant to another.

200. The position of the instantaneous axis with regard to the thrée principal axes, and the angular velocity of rotation, depend on p, q, r, whose determination is very important in these researches; and as they express quantities independent of the situation of the fixed plane roy, they are themselves independent of it.

201. Equations (40) determine the rotation of a solid troubled by the action of foreign forces, as for example, that of the earth when