342 
PROFESSOR DONKIN ON THE 
and the required condition of a maximum will evidently be 
sin /+/?^sin /^=0 (107.) 
adding- the squares of these expressions, we obtain 
(108.) 
which determines the actual value of c; moreover we have 
sini sin sin I _ 
Pi ~~ p ~ 
(109.) 
and it is easy to find (rcos/=j9+jt)^cosI 1 (HO.) 
ff cos t^=p^-\-p cos I i 
so that <7, sin;, sin/^ cos/, cos/^ are all simply expressible in terms p, p^ and I. 
The variation of a is easily found by means of the equation (105.), which gives 
<^<^' = (?/'E;-?E)(Q^-Q). . . . (111.) 
The equation (103.) now gives (see (109.)) 
d ^ 
^(aiicosv— (yoSin»)=^(Q^— Q) ; (112.) 
and from (106.) we obtain 
o-^sin I(/VoCos t'+it/iSin v) = {(jo+p,cos I)E^+(/?^+j!/ cos I)E}(Q^— Q). . (113.) 
The two last equations determine the motion of the principal plane in space, irre- 
spectively of any arbitrary sliding motion which we may attribute to it in its own 
plane. For they give the angular velocities with which it is at any instant moving 
about two lines at right angles to one another in its own plane (see the end of art. 87.). 
They may be put in another form as follows : — the actual value of — H is 
where cos% has the value given above (end of art. 86.); and if the operations indi- 
cated be actually performed, observing that Er=0, E^r=0, &c., and that 
^ d cos Y . 
Ecos%=— ^5 &C., 
the results will be found to be 
( T’ 7* \ , 
^ — ^ lsin(^ — r)sin(0^ — r) 
<f^(/yocosr-l-(yisin{/) = mw^sin X (j»cos(^— v) sin (^^—r)-l-j»;Sin(^—r) cos (^, —{/)). 
Here 0 — v, 0^—v represent, it will be remembered, the angular distances of the planets 
from the line of nodes. 
We will assume for the present the condition ^2=6, so that the plane of xy may 
have no sliding motion, but roll upon the conical surface to which it is always a tan- 
* Referring to the arrangement supposed in the diagram, it will be seen that i becomes negative in the case 
now considered. 
