16 REPORT—1857. 
Sir W. R. Hamilton’s memoir, and relating to the aid to be derived from a 
system of given integral equations (equal in number to the coordinates) in 
the determination of the principal function V. The equations a me’, 
&e. give dV=m(a'dx+y'dy+z'dz), or in the case of a system of points, 
dV = m(a'dx+y'dy+2'dz). If the points, instead of being free, are con- 
nected together by any equations of condition, then, by means of these 
equations, the coordinates 2, y, z of the different points and their differential 
coefficients 2',y',2', can be expressed as functions of a certain number 
of independent variables ¢, y,0, &e., and of their differential coefficients 
¢’, v’,0', &c.; dV then takes the form dV=Xd¢+ Yd)+Zd6+ .. where 
X, Y,Z are functions of ¢,¥,.. ¢', ',... Imagine now a system of in- 
tegrals (one of them the equation of ws viva) equal in number to the 
independent variables ¢, W, 0..; then, by the aid of these equations, ¢’,1)’',’.., 
and, consequently, X, Y,Z.. can be expressed as functions (of the con- 
stants of integration and) of the variables ¢, i, 0,... Hence, attending 
only to the variables, dV= Xdy + Yd+ Zd0+ .. is a differential expression 
involving only the variables ¢, , 0..; but, as Poisson remarks, this expres- 
sion is not in general a complete differential. In the cases in which it is so, 
V can of course be obtained directly by integrating the differential ex- 
pression, viz. the function so obtained is in value, but not in form, Sir W. 
R. Hamilton’s principal function V, for, with him, V is a function of the 
coordinates, and of a particular set of the constants cf integration, viz. the 
constant of vis viva h, and the initial values of the coordinates. Poisson 
adds the very important remark, that V being determined by his process as 
above, then / being the constant of vis viva, and the constants of the other 
given integral equations being e,f, &c., the remaining integrals of the 
problem are * 
dV _ dV _ i aN. 
Fi ae =, dha om 
where r, J, m,.. are new arbitrary constants. But, as before remarked, the 
expression for dV is not always a complete differential. Poisson accord- 
ingly inquires into and determines (but not in a precise form) the condi- 
tions which must be satisfied, in order that the expression in question may 
be a complete differential. He gives, as an example, the case of the motion 
of a body in space under the action of a central force; and, secondly, the 
case considered in Jacobi’s letter of 1836, which he refers to, viz., here 
dV=z'dx+y/'dy, and when the two integral equations are one of them, the 
equation of vis viva £(@?+y")=U+A, and the other of them any integral 
equation a=F (a, y, a’, y') whatever (subject only to the restriction that a 
is not a function of x,y, #+y", the necessity of which is obvious) the con- 
dition is satisfied per se, and, consequently, a'dv+y'dy is a complete 
differential, and its integral gives (in value, although as before remarked 
not iu form) the principal function V; and such value of V gives the two 
integral equations obtained in Jacobi’s letter. 
392. Jacobi’s note of the 29th of November, 1836, ‘On the Calculus of 
Variations, and the Theory of Differential Equations. —The greater part of 
this note relates to the differential equations which occur in the calculus of 
STA ANA 
* Poisson writes = —f+te; there seems to be a mistake as to the sign of / running 
through the memoir. Correcting this, and putting —7 for e, we have the formula oS ttr 
given in the text. 
