ON THE SPECIAL PROBLEMS OF DYNAMICS, 185 
Rotation of a Solid Body ; 
Kinematics of a Solid Body ; 
Miscellaneous Problems. 
As regards the first division of the subject, I remark that the lunar and 
planetary theories, as usually treated, do not (properly speaking) relate to the 
problem of three bodies, but to that of disturbed elliptic motion—a problem 
which is not considered in the present Report. The problem of the spherical 
pendulum is that of a particle moving on a spherical surface ; but, with this 
exception, I do not much consider the motion of a particle on a given curve 
or surface, nor the motion in a resisting medium; what is said on these 
subjects is included under the head Miscellaneous Problems. The first six 
heads relate exclusively, and the head Miscellaneous Problems relates princi- 
pally to the motion of a single particle. As regards the second division of 
the subject, I will only remark that, from its intimate connexion with the 
theory of the motion of a solid body, I have been induced to make a separate 
head of the geometrical subject, “‘ Transformation of Coordinates,” and to treat 
of it in considerable detail. 
I have inserted at the end of the present Report a list of the memoirs and 
works referred to, arranged (not, as in the former Report, in chronological order, 
but) alphabetically according to the authors’ names: those referred to in the 
former Report formed for the purpose thereof a single series, which is not 
here the case. The dates specified are for the most part those on the title- 
page of the volume, being intended to show approximately the date of the 
researches to which they refer, but in some instances a moxe particular speci- 
fication is made. 
I take the opportunity of noticing a serious omission in my former Report, 
yiz., I have not made mention of the elaborate memoir, Ostrogradsky, 
“Mémoire sur les ¢quations différentielles rélatives au probléme des Isopéri- 
métres,” Mem. de St. Pét. t. iv. (6 sér.) pp. 385-517, 1850, which among other 
researches contains, and that in the most general form, the transformation of 
the equations of motion from the Lagrangian to the Hamiltonian form, and 
indeed the transformation of the general isoperimetric system (that is, the 
system arising from any problem in the calculus of variations) to the Hamil- 
tonian form. I remark also, as regards the memoir of Cauchy referred to in 
the note p. 12 as an unpublished memoir of 1831, there is an “ Extrait du 
Mémoire présenté 4 l’Académie de Turin le 11 Oct. 1831,” published in 
_ lithograph under the date Turin, 1832, with an addition dated 6 Mar. 1833. 
The Extract begins thus :—*« § I. Variation des Constantes Arbitraires. Soient 
données entre la variable ¢,. . . m fonctions de ¢ désignées par 2, Ys 2 oija Ch 
autres fonctions de ¢ désignées par u, v, w,. . 2n équations différentielles du 
prémier ordre et de la forme 
dat _ dQ dy dQ dz dQ 
dt ai dl deo diss. ie 
du dQ dvy_ dQ dw dQ &e.” 
ay hes 
dt da’ dt dy” “dé dz 
without explanation as to the origin of these equations; and the formule are 
then given for the variations of the constants in the integrals of the foregoing 
system ; this seems sufficient to establish that Cauchy in the year 1831 was 
familiar with the Hamiltonian form of the equations of motion. 
Bour’s “‘ Mémoire sur l’intégration des équations différentielles de la Mé- 
canique,” as published, Mém. prés, de Inst. t. xiv. pp. 792-821, is substan- 
