DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
99 
we obtain for the final integrals, 
m^rdr { 2mr 2 (h + <p (r ) ) — k 2 } . 
Sh 
sin' 
ctan \ 
v k 1 - 
Let cos / ; then 
k 
-P}“ 2 + sin 
£ 
^- 2 2 = cot /, and the equation (v.) becomes 
• (iv.) 
. (v.) 
. (vi.) 
tan tan i.sin {6— (3), (v.a) 
which expresses that the orbit is in a plane whose inclination to the plane of x, y is i. 
Also (3 is evidently the longitude of the node, reckoned from the axis of x. 
The last term on the left of (vi.) becomes 
Now if £ybe the “argument of latitude” or the angle between the node and the radius 
vector r, we have evidently sin so that the above term is simply S-, and the 
integral (vi.) becomes 
^2mr 2 (h-\-<p(r)) — /c 2 | ' (vi.a) 
30. To apply the above expressions to the case of the undisturbed motion of a 
planet, we have only to put p(r) = — > where m is now the mass of the planet, and ^ 
the sum of the masses of the sun and planet, the origin of coordinates being placed 
at the sun. It would be useless to give the well-known expressions to which the in- 
tegrations now lead, my object being merely to obtain a set of normal elements. Now 
in this case we have (by well-known theorems), if a be the semiaxis major, e the 
excentricity, and / as before the inclination, 
h =~2d’ *= *Vz(l — e 2 ), 
and therefore c— v' J a,a(l — e 2 ) .cos /. 
Also, if we take for the inferior limit of the integrations in (iv.) and (vi.a) the 
minimum value of r, or the perihelion distance, it is plain that « will be the longitude 
of the node, reckoned from the perihelion in the plane of the orbit, and — r the time 
of perihelion passage. Thus we have the following six elements, arranged in con- 
jugate pairs : — 
—(time of perihelion passage) 
V pa(\— e 2 ), (angle between node and perihelion) 
v' H>a{ 1 — e 1 ) .cos /, (longitude of node). 
o 2 
