> ig Et Fe 
ON THEORETICAL DYNAMICS. 19 
AVG) sep gM cee MA sili | 
Be gp Gs Mr 
By op ig AMeay AN 
Bipir ida, oyaB, Ae 
where 7, A, p are new arbitrary constants. 
36. Jacobi’s principal function V.—Secondly, when the equation of vis 
viva is not satisfied. Here U contains the time ¢ and we have no such 
equation as T=U-+A, but along with the coordinates 2, y, z there is intro- 
duced a new variable H, and V is defined to be a complete integral of the 
partial differential equation 
1 dV\? (dV\2, (dV\2 
ati) +g +(e) }=U+H: 
where, in the expression for U, it is assumed that¢ is replaced by — qi There 
are, consequently, four independent variables, and a complete solution must 
contain, exclusively of the constant attached to V by mere addition, and 
which disappears from the differential coefficients, three arbitrary constants 
a,(,y- The integral equations are shown to be 
dx 
Mia 10 Ab ean AVL 
Gee) 26 Maco baoetes 
ey 2) 
Ts Sih 
where 2, p, v are arbitrary constants, viz., eliminating H from the first 
three equations by the assistance of the last equation, we have the inter- 
mediate integrals ; and eliminating H from the second three equations by 
the assistance of the last equation, we have the final integrals. The substi- 
tution of the above values a, &c., in the partial differential equation gives 
i 
T=U+H, that is, H(=T—U) is that function which, when the condition 
of vis viva is satisfied, becomes equal to f, the constant of vis viva. 
Jacobi’s extension of the theory to the case where the condition of wis 
viva is not satisfied, appears to have attracted very little attention; it is 
indeed true, as will be noticed in the sequel, that this general case can be 
reduced to the particular one in which the condition of vis viva is satisfied, 
but there is not it would seem any advantage in making this reduction; 
the formule for the general case are at least quite as elegant as those for 
the particular case. 
37. Jacobi, after considering some particular dynamical applications, pro- 
ceeds to apply the theory developed in the first part of the memoir to the 
, general subject of partial differential equations; the differential equations 
of a dynamical problem lead to a partial differential equation, a complete 
solution of which gives the integral equations. Conversely, the integral 
equations give the complete solution of the partial differential equation, and 
applying similar considerations to any partial differential equation of the 
first order whatever, it is shown (what, but for Cauchy’s memoir of 1819, 
which Jacobi was not acquainted with*, would have been a new theorem) 
~ * Jacobi refers to Lagrange’s ‘ Lecons sur la Théorie des Fonctions,’ and to a memoir by 
Pfaff in the ‘ Berlin Transactions’ for 1814, as containing, so far as he was aware, every- 
c2 
