8 REPORT—1857. 
the. coordinates ¢,..., but he does not complete the transformation of the 
differential equations by the introduction therein of the new variables s,... 
in the place of the differential coefficients ¢',..; this very important trans- 
formation was only effected a considerable time afterwards by Sir W. R. 
Hamilton. Poisson then assumes that the undisturbed equations are in- 
tegrated in the form above adverted to, viz., that the several elements a, b.. 
are given as functions of the time ¢, and of the coordinates ¢, &c., and their 
differential coefficients ¢', &c., or what is the form ultimately assumed, as 
functions of the time ¢, of the coordinates ¢,.., and of the new varia- 
bles s, &c.; and he then forms the functions 
__0(a, b) 
ear Ser 
0(a,b)_dadb_dbda 
0(s,¢) dsdp dsdo 
(the notation is the abbreviated one before referred to), and he proves by 
differentiation that the differential coefficient of (a, ) with respect to the time 
vanishes : that is, that (a, 6) which, by its definition is given as a function of ¢ 
and of the coordinates ¢,..., and of the new variables s,..., is really a 
constant. Upon which Poisson remarks—“ On concoit que la constante... 
sera en général une fonction de a et & et des constantes arbitraires con- 
tenues dans les autres intégrales des équations du mouvement; quelquefois 
il pourra arriver que sa valeur ne renferme ni la constante @ ni la con- 
stante 6; dans d’autres cas elle ne contiendra aucune constante arbitraire, et 
se réduira a une constante déterminée ; mais afin,” &c. 
13. The importance of the remark seems to have been overlooked until 
the attention of geometers was called to it by Jacobi; it has since been 
developed by Bertrand and Bour. 
It is clear from the definition that (a,b)=—(d,a). It may be as well to 
remark that the denominator of the functional symbok is (s, ¢) and not 
(9, s), which would reverse the sign. 
14. The equations for the variations of the elements are without difficulty 
shown to be da dQ, 
ca fe) ee Ei 
gem ee agi 
ard 
where 
which have the advantage over those of Lagrange of giving directly =4 &e. 
in terms of — &c., instead of these expressions having to be determined 
a 
from the value of dQ, &c., in terms of da &e. 
da dt 
15. Poisson applies his formule to the case of a body acted upon by a 
central force varying as any function of the distance, and also to the case 
of a solid body revolving round a fixed point. There is, as Poisson remarks, 
a complete similarity between the formule for these apparently very dif- 
ferent problems, but this arises from the analogy which exists between the 
arbitrary constants chosen in the memoir for the two problems. The 
formulz obtained form a very simple and elegant system, and one which, 
although not actually of the canonical form (the meaning of the term will 
be presently explained), might by a slight change be reduced to that form. 
16. I may notice here a problem suggested by Poisson in a report to the 
Institute in the year 1830, on a manuscript work by Ostrogradsky on 
