DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
341 
similarly, we find 
/?,sin I./=cos I.E^Q^+EQ,— (a/ocos sinv) sin I ; 
and since 1'=;'—/, we obtain from these 
/?/>^sin I.r=jo(cos I.Ep^+EQ^)+jt?j(cos I.EQ + E;^), . . . (104.) 
an equation which may be transformed as follows : — since 
1=0, 1=0 (art. 79), 
„ , i, I d£l . d£l 
we find trom art. 55, p =-^+^ = E12, 
and similarly p\=.Efl^, whence it is easily seen that the above equation may be written 
in the form 
(T^jo^cos I)'+7?EQ^+j9^Ep = 0 (105.) 
If we investigate, from the above expressions for / and /, the values of {p sin /)' and 
, (PiSinti)', we find 
sin I.(j» sin /)'= — (cos /^.E-f-cos /.E^)Q — p cos i sin I(<y(,cos t'+a^i sin v), 
sin l.(/>^sin /^)'=(cos /.E^+cos /^.E)Q^— ja^cos qsin I(a;oCosyd-a/,sin v), 
and, adding these equations, 
sin \ .{p sin /+jo^sin q)'=(cos /^.E4-cos/.E;)(Q,— Q) 
— (jocos/+p,cos/;)sinl.(iyocosv+(yisint'). . . . (106.) 
With respect to this equation and (103.), it may be observed that iwocosv+(i»isin v 
is evidently the angular velocity of the plane of xy about the line of nodes, and 
a;iCos v— ft/flSin I' is the angular velocity of the same plane about a line in itself per- 
pendicular to the line of nodes ; or, which comes to the same thing, the angular 
velocity of the line of nodes itself in fixed space, estimated perpendicularly to the plane 
of xy. 
88. The position of the plane of xy at any instant has been so far left arbitrary, ex- 
cept that it has been subjected to the condition oi passing through the line of nodes. 
As a further assumption, that which most naturally presents itself is, that the plane 
of xy should always coincide with the principal plane. By the principal plane I mean, 
of course, that on w hich at any instant the sum of the projections of the areal velo- 
cities (multiplied by the masses) of the two planets about the sun, is a maximum ; it 
evidently always passes through the line of nodes, and would be the invariable plane 
if the disturbing functions vanished. 
To determine the position of this plane we have (see art. 80.) to express the con- 
dition that < and are always so taken, subject to the equation tliat the ex- 
pression m\/ja,a(l —e^) .cos —e)^). cos shall be a maximum. We will 
put ff for the value of this expression, so that using p and p^ with the same meaning 
as before (see art. 86.), we have 
a—p cos /+jo,cos 
2 z 2 
