198 
MR. IVORY ON THE THEORY 
But, if d t represent the element of the time supposed to flow uniformly, the 
• • • • • • d x dv dz 
actual velocities with which the coordinates increase are, -j- ; and the 
„ , , . . ddxddy ddz , „ 
increments of these velocities, -j^r, -j-jz, are the effects produced by all 
the forces that urge the planet. Equating now the forces really in action to 
the measure of the effects they produce, and observing that the two equivalent 
quantities have been estimated in opposite directions, we obtain the following 
equations for determining the place of P relatively to S at any proposed instant 
of time, 
d d x , [x x m' [x' — x) ml x' 
TF + 73 ~ f ~ ~? r> 
ddy \xjy^ nl (y' — y) _ m’y' 
dt 2 r 3 g 3 r 13 ’ 
ddz fj, z m! [z! — z) irl z' 
d t 2 r 3 g 3 r 13 
If we now assume 
_ m' 
R = ^ x 
f 
f \l (x 1 — x) 2 + (y 1 — y) 2 + ( z 1 — z) 2 
x x 1 + yi/ + z z' 1 
{at* + y' 2 + z' 2 f J 5 
it will be found that the partial differentials, , ih X ^ x jy, 
d. R 
f* X j 7> are 
respectively equal to the quantities on the right sides of the last equations, 
that is, to the disturbing forces tending to increase the coordinates x, y , z. 
These equations may therefore be thus written. 
d dx x d R 
jx d t 2 ' r 3 d x ’ 
d dy y JR 
\xdt 2 ' r 3 dy ’ 
ddz z JR 
ixdt 2 ' I s ~ d z’ 
> 
(A) 
If it be asked, What notion must be affixed to the symbol y>dt 2 }, it will be 
recollected that ^ is the attraction between S and P at the distance r ; and if 
we suppose that P describes a circle, of which unit is the radius, round S, the 
centripetal force in the circle will be or (x ; and the velocity with which P 
