ON THEORETICAL DYNAMICS, 17 
- variations, including, indeed, the differential equations of dynamics, but which 
belong to a different field of investigation. The latter part of the note relates 
more immediately to the differential equations of dynamics. ‘The author re- 
marks, that, in any dynamical problem of the motion of a single particle for 
which the principle of vzs viva holds good, if, besides the integral of vis-viva, 
there is given any other integral, the problem is reducible to the integration 
of an ordinary differential equation of two variables, and that it is always 
possible to integrate this equation, or at least discover by a precise and 
general rule the factor which renders it integrable. This would seem to 
refer to Jacobi’s researches on the theory of the ultimate multiplier, but the 
author goes on to refer to a preceding communication to the Academy of 
Paris (the before-mentioned letter of 1836), which does not belong (or, at 
least, does not obviously belong) to this theory. He speaks also of a class 
of dynamical problems, viz. that of the motion of a system of bodies which 
mutually attract each other, and which may besides be acted upon by 
forces in parallel lines, or directed to fixed centres, or even to centres the 
motion of which is given; and, he remarks, in the solution of such a pro- 
blem, the system of differential equations being in the first instance of the 
order 2n (that is, being a system admitting of 2x arbitrary constants), then 
if one integral is known, it is possible by a proper choice of the quantities 
selected for variables to reduce the system to the order 2n—2. If another 
integral is known, the equation may in like manner be reduced to a system 
of the order 2n—4, and so on until there are no more equations to be in- 
tegrated ; and thus the operations to be effected depend only upon quadra- 
tures. All this seems to refer to researches of Jacobi, which, so far as I 
am aware, have not hitherto been published. The results correspond with 
those recently obtained by Bour, post, Nos. 66 and 67. 
33. Jacobi’s memoir of 1837.—Jacobi refers to the memoirs of Sir 
W. R. Hamilton, and he reproduces, in a slightly different form, the inves- 
tigation of the fundamental property of the principal function S. The case 
considered is that of a system of m particles, the coordinates of which are 
connected together by any number of equations; but it will -be sufficient 
here to attend to the case of a single free particle. The equations of motion 
are assumed to be : 
mee _ dU nit= dU ma? _ aU 
de ~ da? de ~ dy’ "de de” 
But U is considered as being a function of 2, y, z and of the time ¢, that is, 
it is assumed that the condition of wis viva is not of necessity satisfied. The 
definition of the function S is 
t 
s+f [U+ame+y* +e") ]ae which, 
0 
when the equation of vis viva is satisfied, that is, when T=3m(x?+y"+42!?) 
=U-+A, agrees with Sir W. R. Hamilton’s defiuition S=2f" Udi+ht. The 
function S is considered as being, by means of the integral equations as- 
sumed as known, expressed as a function of ¢, of the coordinates x, y, z, and 
of their initial values a,b,c. And then it is shown that S satisfies the equa- 
tions 
dS , aS , aS 
dee gm > aie 
dS dS mb dS 
Ge eb 7 mes 
1857. ¢c 
