OF THE TRANSLATION OF A DIRECTED MAGNITUDE. 
187 
m fdrJ du' du du\ 
* 2\dt ^ dt dt^ dt)' 
Here denotes the square of the velocity (art. 49), and thus the expression 
denotes the ordinary vis-viva. 
(81.) Description of Areas . — It is clear that -^u.du is the area described in the time 
dthy U-, for it is the half parallelogram formed on u and du. But it is to be noticed 
that \u.du represents this area, not only in magnitude, but also in position. 
If we put ^u.du=Kdt, we find 
whence 
^_l 
dt “2 
du du'' 
dt dt j 
\ . du du „ , 
(art. 32); 
dh. 1 d^u 
'di~'^’lt^' 
( 1 .) 
A, here, is a symbol which represents two important things: — 1st, in magnitude, it 
is the ordinary rate of description of area ; 2ndly, its plane is the plane con- 
taining the radius vector and the direction of motion, i. e. the plane of the orbit of m. 
DA is the' symbol of a line perpendicular to the plane of the orbit, and equal in 
magnitude to the rate of description of area. 
If we put u=xcc-\-yi^-\-zy, and perform the operation indicated in equation (1.), we 
find. 
^^(DA) 
dt 
/ d^i! (Px\ / d'^z 
{^dF-^dFj'y+'ym- 
(82.) Expression for Effective Force with reference to radius vector, angular velocity, 
and plane of orbit . — Let r denote the magnitude and a the ^‘direction" of m; let |3 
denote a direction in the plane of the orbit of m (and, of course, at right angles to 
a) ; then y will be a direction always perpendicular to the plane of the orbit : lastly, 
let a denote the angular velocity of u, and cJ the angular velocity of the plane of the 
orbit about u, a and J being numerical quantities. On referring to art. 19, the 
following relations are manifest, 
dec d^ 
“ = dt=^^ 
for ^ and ^ denote, in magnitude and direction, the velo- 
cities of the extremities of the directed units a and (3, and 
the figure will show what their vnlues are in terms of &> 
and i>'. 
Fig. 35. 
* j/du du\ dhi du , du d^u 
dt ) dt dt dt dt 
= 2^—xdu (art. 33). 
dt^ 
2 B 2 
i / 
— =r y 
dt 
