ON THKORETICAL DYNAMICS. 9 
Celestial Mechanics, viz., in the case of a body acted upon by a central force, 
the effect of a disturbing function, which is a function only of the distance 
from the centre, is merely to alter the amount of the central force ; and the 
expressions for the variations of the elements should therefore, in the case 
in question, admit of exact integration ; the report is to be found in Crelle, 
t. vil. pp. 97-101. 
17. The two memoirs of Lagrange and Poisson, which have been con- 
sidered, establish the general theory of the variation of the arbitrary con- 
stants, and there is not, I think, very much added to them by Lagrange’s 
memoir of 1810, the second edition of the ‘ Mécanique Analytique, 1811, 
_or Poisson’s memoir of 1816. The memoir by Maurice, in 1844, belongs to 
this part of the subject, and as its title imports, it is in fact a development 
of the theories of Lagrange and Poisson. 
18. There is, however, one important point which requires to be adverted 
to. Lagrange, in the memoir of 1810, and the second edition of the 
‘Mécanique Analytique,’ remarks, that for a particular system of arbitrary 
constants, viz., if a,... denote the initial values of the coordinates &,.. and 
A,.. denote the initial values ‘of ae . then the equations for the varia- 
tions of the elements take the very simple form 
da__ dQ dr_da 
dine Wei’, aie ida 1 
This is, in fact, the original idea and simplest example of a system of can- 
onical elements ; viz. of a system composed of pairs of elements, a, A, the 
variations of which are given in the form just mentioned. 
19. The ‘ Avertissement’ to the second edition of the ‘Mécanique Analytique’ 
contains the remark, that it is not necessary that the disturbing function 
dQ, dQ dQ 
We) dytiae 
symbols standing for forces X, Y,Z, not the differential coefficients of one and 
should actually exist ; may be considered as mere conventional 
the same function, and then = will be a conventional symbol standing for 
a 
dQ dx , dQ, dy , dQ dz 
dx da dyda dz da@ 
the formule will subsist as in the case of an actually existing disturbing 
function. 
20. Cauchy, in a note in the ‘ Bulletin de la Société Philomatique’ for 
1819 (reproduced in the ‘Mémoire sur l'Intégration des Equations aux 
Derivées Partielles du Premier Ordre,’ ‘ Exer. d’ Anal. et de Physique Math.,’ 
t. ii. pp. 238-272 (1841)), showed that the integration of a partial dif- 
ferential equation of the first order could be reduced to that of a single 
system of ordinary differential equations. A particular case of this general 
theorem was afterwards obtained by Jacobi in the course of his investiga- 
tions (founded on those of Sir W. R. Hamilton) on the equations of dyna- 
mics, and he was thence led to a slightly different form of the general 
theorem previously established by Cauchy, viz., Cauchy’s method gives the 
general, Jacobi's the complete integral, of the partial differential equation. 
The investigations of the geometers who have written on the theory of * 
dynamics are based upon those of Sir W. R. Hamilton and Jacobi, and it is 
therefore unnecessary, in the present report, to advert more particularly to 
Cauchy’s very important discovery. 
_21. I come now to Sir W. R. Hamilton’s memoirs of 1834 and 1835, 
which are the commencement of a second period in the history of the sub- 
and similarly for ae &e.; and this being so, all 
