DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
337 
Application to the Problem, of Three Bodies. 
84. I propose to exemplify the preceding- principles by applying them to the trans- 
formation of the equations which determine the motion of three mutually attracting 
bodies, considered as material points. Let them be called the sun and two planets, 
and let the origin of coordinates be placed at the sun, and the notation be the same 
as in arts. 77-79, so that M, m, m^, are the three masses, M.-\-m=pj, &c. 
Let the elements be chosen as in art. 55, and longitudes be measured, as before, 
along the plane oi xy (from the axis of x) as far as the node, and then along the plane 
of the orbit. 
Putting for convenience wrR=n, we have 
n = mm^ { {P + r^ — 2rr^ cos * — rr~^ cos %; } 
2rr^cos 7jr“^cos %}, 
where % is the angle between the two radii vectores. Let I be the mutual inclina- 
tion of the planes of the two orbits, and let the angular distances, on the planes of 
the orbits, between their ascending nodes on the plane of xy and their line of inter- 
section, be respectively v, ; so that & — v — v, are the angular distances of 
the two radii vectores from the line of intersection (which we may call, simply, the 
line of nodes). We shall then have 
cos %=cos(^— y — {')cos(^,— y^ — y^)-l-sin(^— y— j')sin(^^— y^ — y^)cosI'| ^ 
cosI = cos / cos /^-j-sin / sin i 
and y, y, are functions of /, v^—v, determined by the equations' 
cotysin(i/^ — v) = — cot/^sin / + cos(f,— yjcos ^ 
cot y^sin(i'^— i') = cot ; sin /^+cos(j'^— >^) cos J 
Now considering Cl as expressed, on the one hand, in terms of r, 6, 0^, v, and 
on the other in terms of all the elements and t, we have rigorously, as may easily be 
proved in the usual way-f-, 
dCl dCi dCl dCl 
dS ds ' dtjs^ d^i dsid-sr’ 
* The arrangement referred to here and in the following 
articles will be made clear by the accompanying diagram. 
f Since the values of r and 9 in terms of the elements and 
t are of the forms 
T =f{fndt -)- £ — ot), Q=fndt-\-s->r <p{fndt -j- £ — ra-) , 
we have 
ds d-m ’ ds dvr ’ 
from which the equations in the text follow immediately. It 
may be as well to remind the reader who may happen to 
recollect the note to the Astronomer Royal’s tract on the 
Planetary Theory (p. 91, ed. 1831}, that that note refers to 
a different way of measuring longitudes. 
