236 REPORT—1862. 
dé oS 
eect Le ae elt Be | — 
Vise ae ? o that writing —=am u, 
the integral equation is nt—e=vw, or u is an angle varying directly as the 
time (and corresponding to the mean longitude, or, if we please, to the mean 
anomaly in the problem of elliptic motion). And then p, q, r are expressed 
as elliptic functions of vu. The value of the modulus &, and that of 
n (nt—e=u ut supra) are 
pe Mee nS 
c ABC ? 
jas (A—B)(P—Ch) 
ABC ; 
p= digufetath, cos am w, 
fh 
o== — 
Y BLB_C sll am wu, 
—P+Ah 
cC.A—C 
187. Substituting for p, g, r their values in terms of u, we have dé, and 
thence @ (the longitude of the node of the equator on the invariable plane) in 
the form 
pression for dt takes the form ndi= 
and then 
= Aam wu, 
1 , : a | 
0=—7z, u+ill(u, ia) (i= —1), 
which by Jacobi’s formule for the transformation of the elliptic integral of 
the third class becomes 
lores -7 O(u—ar) 
=] —— Zi i ee 
. ( resp (ai) ut 3 log @(u-paiy 
which Rueb reduces to the real fc_ 
6=—n'u+tan-! W, 
W being given in the form of a fraction, the numerator and denominator 
whereof are series in multiple sines and multiple cosines respectively of 
mu 
188. Rueb investigates also the values in terms of u of the cosine inclina- 
tions of the instantaneous axis to the axes fixed in space; and he obtains a 
very elegant expression for the angle ¢, which is the angular distance from « 
of the projection on the plane of wy (the invariable plane) of the instantaneous 
axis; viz., this is 
g=tan( ABn A amu a : 
~ (A—B)/ sin am w cos am wu 
and there is throughout a careful discussion of the geometrical signification 
of the results. 
189. The advance made was enormous; the result is that we have in terms 
of the time sinc sin ¢, cost sin ¢, cos ¢ (the cosine inclinations of the inva- 
riable axis to the principal axes), and also 0, the longitude of the node. The 
cosine inclinations of the axes of x and y to the principal axes could of course 
be obtained from these, but they would be of a very complicated and un- 
ti 
