DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
347 
pp, sm I.r=;>,|cos l( * +~d;+i^] 
+;,{cosl(^+§;)+f+§}. 
94. The only parts of the preceding- expressions of which the deduction is not per- 
fectly obvious, are the terms involving- I in the values of s', s', vif', ss. They are 
obtained, as has been sufficiently explained, from the expressions in art. 55, to which 
are to be added the values of &c. ((95.), art. 79.) ; on putting- y=0, disappears 
from the latter; and the values of cos/, cos/,, tan^? tan^ are to be introduced 
2 2 
(equations ( 110 .), ( 112 .), ( 116 .), ( 117 .)). After some rather troublesome reductions, 
the expressions above given will be found. 
In these equations it will be recollected that the mean longitude of m is represented 
c* . . dn 
by j ndi-\-s, and the differentiation with respect to a in ^ is only performed so far 
as a appears explicitly. If we wished that the mean longitude should be expressed 
by nt-\-s, the only change in the equations would be that the differentiation with 
respect to a must be total ; i. e. must extend to a as contained in w. A similar 
remark applies of course to s,. 
In actual use it would be more convenient to introduce R, R, instead of Q, Q, ; the 
latter functions give a rather more symmetrical form to the equations, and are more 
convenient in general investigations. (The relation between them here is merely 
Q=»^R, Q,=m,R, ; in another part of the paper the symbol Q was used for — /«R 
(art. 55.)). 
95. If the equations of art. 93. were completely integrated, the intrinsic motion 
of the system would be completely determined ; that is, we should know at any 
instant the dimensions of the two orbits, the mutual inclination of their planes, the 
position of their axes with respect to the line of nodes, the place of each planet in its 
orbit, and (by (1 10.)) the inclination of each orbit to the principal plane. 
The position of the system relatively to fixed space would then have to be sepa- 
rately determined as follows : — 
The three quantities co^, co., (see end of art. 92.), of which the values are ((112.), 
(113.), (116.)) given by the equations 
sinI.».=|(p+p,cos I)(|-+^) + (ft+?cosI)(^5+^)|(£2-Q) 
ffsin I./y2= 
^'4- cos I 
p 
\d£l (p 
3 A 
MDCCCLV. 
