OF THE TRANSLATION OF A DIRECTED MAGNITUDE. 
203 
Now let us take new directions {od, |3', y') so, that y' shall coincide for a moment with 
the polar axis y, while the plane (a'y) contains the sun’s distance v! ; then 
u'=x'ot!-\-z'y 
Did .y ~ — 
wherefore (14.) becomes 
The coeflScient of (3' here, being multiplied by X, may be regarded as invariable 
during one day, inasmuch as x’ and z' take a j-ear to go through their values : also, since 
I implies that the earth is considered as fixed, the sun, and therefore the direction |3', 
must be supposed to revolve about y from east to west, through 360° in the day. 
The velocity ^ therefore is constant in magnitude but changes its direction (which 
is always perpendicular to y) uniformly through 360° in the day. It appears there- 
fore that the point {oj) describes a daily circle, and therefore the line a describes a 
daily cone about y. From this it follows (observing that co and y make a very small 
angle with each other), that the mean daily angular motion in space of y and that of 
the direction of a (manifestly ^ very nearly) are identical. Wherefore 
dy 
dt 
\ dm X Srrd 
n dt n r'^ 
z'Du'.y, by (13.) ; 
or, since 
z'=u'xy, 
w=s^(“'x5')D“'-y (>5-) 
If now we assume y' to be at right angles to the plane of the ecliptic, and cd to 
point towards the first point of Aries, we have 
id—r^iod cos w'^+(3' sin /i7), 
7f^ 
where rdt is the sun’s longitude, being 
Wherefore, observing that y is at right angles to a', we find 
^_3^(yX^,) «7Da'.y+sin w7Df3'.y). 
STT • a • 
If we integrate this between the limits 0 and y, we find the annual variation of y, 
which therefore is 
n 
(yX^')I^f3'.y. 
This represents in magnitude and direction the actual space described by the 
point (y) in one year, i. e. the angular motion of the pole, or the precession. If sr 
denote the obliquity of the ecliptic, 
y = y’ COS ro-+|3' sin 
2 D 2 
