10 REPORT—1857. 
ject. The title of the first memoir shows the object which the author pro- 
posed to himself, viz., the discovery of a function by means of which the 
integral equations can be all of them actually represented. The method 
given for the determination of this function, or rather of each of the several 
functions which answer the purpose, presupposes the knowledge of the 
integral equations ; it is therefore not a method of integration, but a theory 
of the representation of the integral equations assumed to be known. I 
venture to dissent from what appears to have been Jacobi’s opinion, that 
the author missed the true application of his discovery ; it seems to me, that 
Jacobi’s investigations were rather a theory collateral to, and historically 
arising out of the Hamiltonian theory, than the course of development: 
which was of necessity to be given to such theory. But the new form ob- 
tained in Sir W. R. Hamilton’s memoirs for the equations of motion, is a 
result of not less importance than that which was the professed object of 
the memoirs. 
22. Hamilton’s principal function V.—The formule are given for the 
case of any number of free particles, but, for simplicity, I take the case of a 
single particle. The equations of motion are taken to be 
d'z_dU, 
df dz 
Py a 
d@ dy’ 
@y_ dU. 
"de dz 
so that the vis viva function is 
T=im(e"+y"+<"), 
and the force function, taken with Lagrange’s sign, would be —U. It is 
assumed that the condition of vis viva holds, that is, that U is a function of 
a, y,z only. The initial values of the coordinates are denoted by a, b,c, 
and those of the velocities by a/,b',c'. The equation of vis viva is 
T=U+H, and this gives rise to an equation Tj=U,)+H of the same form 
for the initial values of the coordinates. The author then writes 
t 
v=( 2Tdt, 
0 
an equation, the form of which implies that T is expressed as a function 
of the time and of the constants of integration a,b, ¢, a’, b',c'. The method 
of the calculus of variations leads to the equation 
SV=m(a' dx +y'dy + 262) —m(a'da +-b'tb + c'de) + 08H, 
to understand which, it should be remarked that the coordinates a, y, 2, 
and the velocities 2’, y', z', being functions of ¢ and of a, b, ¢, a’, b', e', then 
V is, in the first instance, given as a function of these quantities. But 
“x,y,z being functions of a, b,c, a', b',c',t, we may conversely consider 
a',b',c' as functions of a, y,z, a, 6,e,t, and thus V becomes a function of 
X,Y, 2, a,b,c, In like manner H is a function of a, y, 2, a, 6b, ¢,é, and, 
eliminating ¢, we have V a function of 2, y, 2, a,b,c, H, which is the form 
in which in the last equation V is considered to be expressed. The equa- 
tion then gives 
