DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC, 
343 
gent plane. If, then, we call a the actual angular velocity of the principal plane 
about its instantaneous axis of rotation, so that and put j for the angle 
between the line of nodes and the instantaneous axis (which is the line in which the 
principal plane is intersected by its consecutive), we shall have 
i!yoCOSt'-[-(y,sini'=iy cosj and cos > — a/oSin sinj ; 
and the above equations give 
a cotj—p oot(^— v)-j-jt?^cot {6—v), 
a result which may also be put in the form (see (109.)) 
sin I cotj=sin /^cot (^— ;') — sin / cot (^^— j/). 
It is very easy to show by spherical trigonometry that this equation signifies that 
the instantaneous axis coincides with the line in which the principal plane is inter- 
sected by the plane of the two radii vectores*. In other words, we have this theorem : 
The principal plane always turns about the line in which it is intersected by the plane 
of the two radii vectores. 
It follows of course that the principal plane always touches the conical surface 
described in fixed space (relatively to the sun) by the said line. I think it probable 
that most persons would expect, at first sight, that the principal plane would always 
touch the conical surface described by the line of nodes, which, as has been just 
shown, is not the case. It is perhaps worth while to verify this result by inde- 
pendent reasoning. 
89. Let Roman letters refer, as in former articles, to axes of coordinates having any 
arbitrary fixed directions. Then, putting A=:m{yz'—zy'), &c. (as in art. 80.), and 
using I, 7], ^ for current coordinates, the equation to the principal plane is 
(A+A,)H-(B-l-B>-f(C+CX=0; (114.) 
and the line in which this plane is cut by its consecutive is determined by combining 
the equation (1 14.) with that obtained from it by differentiating the coefficients of 
I, 7^, ^ with respect to t ; namely, 
(A'H-A;)|-f (B'+B>+(C-f CX=0 (1 15.) 
Now from the fundamental equations 
tnx”+mfz^=^> See,, 
we obtain A' = m(yz"—zy")=y^^—z^ = wm^(yz^—zy, )(§-' — r”') 
* Let if/ be the angle between the principal plane and the plane of the radii vectores ; the former plane 
divides the angle I into two parts, of which one is and the other is — i ; and we get two spherical triangles 
which have a common side j, with adjacent angles and ip in one, — ( and tt— y in the other; hence 
cot (9 — »')sin y=cot\{/ sin i -i- cos j cos i 
cot($^ — y)sinj=cotil/ sin cos J cos 
and if cot ip be eliminated between these, the result is the equation in the text. 
