ON THE SPECIAL PROBLEMS OF DYNAMICS. 239 
-, Hu k He _ Ou 
Vi sin amu= Ou? Jt cos amu, VE Aamu= On’ 
and then considering the cosine inclinations of the principal axes to the 
invariable axis and the revolying axes in the inyariable plane, these are 
all fractions which, neglecting constant factors, have the common deno- 
minator Qu; a’, 8’, y' (as shown by Rueb’s formule) have the numerators 
Hu, Hu, and ©,w respectively; and a, a have the numerators H (w+ia) 
+H (u—ia), B, 3’ the numerators H, (wu—7ta) +H, (w+ia), y, y' the nume- 
rators © (w+ia)+0 (w—ia) “respectively: there are also expressions of a 
similar form for the angular velocities about the axes of a and y; that about 
the axis of z (the invariable axis) haying, as was known, the constant value 
. The memoir is also very valuable analytically, as completing the systems 
of formuls given in the “ Fundamenta Nova” in reference to elliptic integrals 
of the third class. 
199. It is worth noticing how the results connect themselves with Poinsot’s 
theorem of the rolling and sliding cone, the velocity of the rolling motion 
depends only upon the position, on the cone, of the line of contact, so that 
the same series of velocities recur after any number of complete revolutions 
of the cone; that is, the total angle described by the line of contact in conse- 
quence of the rolling motion, consists of a part varying directly with the 
time (or say varying as w) and a periodic part; the former part combines 
with the similar term arising from the sliding motion, and the two together 
give Jacobi’s term nw. } 
200. Somoff’s memoir (1851), written after Jacobi’s Note in the ‘ Comptes 
Rendus,’ but before the appearance of the memoir in Crelle, gives the de- 
monstration of the greater part of Jacobi’s results. 
201. Booth’s “Theory of Elliptic Integrals &c.” (1851) (contemporaneous 
with the publication of Poinsot’s memoir of 1834) contains various interest- 
ing analytical developments, and, as an interpretation of them, the author: 
obtains (p. 93) the theorem of the rolling and sliding cone. The investiga- 
tions inyolye the elliptic integrals, but not the elliptic or Jacobian functions. 
202. Richelot’s two Notes (Crelle, tt. xlii. & xliv.) relate to the solution 
of the problem of rotation given in his memoir “Eine neue Losung &c.” 
(1851). This is an application of Jacobi’s theorem for the integration of a 
system of dynamical equations by means of the principal function S (see my 
“Report” of 1857, art. 34). Retaining Richelot’s letters ¢, y, 0, which 
signify 
i, the longitude of the node, 
0, the inclination, 
, the hour-angle, 
the question is to find a complete solution of the partial differential equation 
_1f(av dV\' sing dV F 
0-551 (% cos 0455) sin @ do cos of 
» 
1 { (dV dV\ cos¢ dV. ; 
' ton (ap cos +57) seat ay 80 0 
1 /dV\? dV 
+26 (as) tat} 
that is, a solution inyolying (besides the constant attached to V by a mere 
