340 
PROFESSOR DONKIN ON THE 
The variations of the elements will now be found by introducing the above values of 
d£L dSli dn d£l^ 
dv dv^ di dll 
in the expressions given in art. 55, and completing these expressions by the addition 
of the terms (95.), art. 79. But the angular velocities Mq, are no longer wholly 
arbitrary, since we have made one assumption concerning the motion of the axes, 
which implies one relation between these quantities and the elements. In order to 
determine completely, it will be necessary to make two more assumptions ; 
but first we will investigate the relation already implied. 
86. The complete expressions for 7, 7, obtained in the way mentioned in the last 
article, from arts. 55 and 79, may be written in the following form : put for brevity 
and put for n, and jM- for ; then 
p sin 
1 
1 , 
- cot /.((Vo Sin cosv )—&)2 
dSl/ 
'PiSinti dif 
-j- cot sin v—coi cos J';) — <^2 
( 102 .) 
and if the latter equation be subtracted from the former, and the conditions 
^ ^ dD, dn d^li dfli ^ ^ j 
di 
dl di 
dl^ 
be introduced, the result is easily found to be 
(iVo sin v—a)i cos i')sin T 
sin I dD,i sin dD, 
(103.) 
Pi dl' p dl 
This is the relation between <^1 nnd the elements and t, implied by the one 
assumed condition that the plane of passes through the line of nodes. The angular 
velocity of the system of axes about the axis of x, is so far left, as it evidently 
ought to be,, perfectly arbitrary. 
Q and are now functions of the following elements* only : — 
And we now have 
cos %= cos {6—v) cos (^ 1 — 1 ')+ sin (0—v) sin (O—v) cos I. 
87. The complete expression for /, derived from arts. 55 and 79, is easily put in 
the form 
' — + ( 1 - cos i) EQ J - (a;oCos 1-+^. sin p ) ; 
and, on introducing the assumptions of the preceding articles, this will be found to 
become, after simple reductions, 
/? sin I./= — (cos I.EQ + E^Q) — (a^oCos v+iWiSin (')josinl ; 
* On the diflference between (e) and e see above, art. 55. We cannot strictly call a function of a and s. 
