18 REPORT—1857. 
so that the intermediate and final integrals are expressed by means of the 
principal function 8S. 
~ 34, But Jacobi proceeds, “the definition assumes the integration of the 
differential equations of the problem. The results, therefore, are only 
interésting in so far as they have reduced the system of integral equations 
into a remarkable form. We may, however, define the function S in a 
quite different and very much more general manner.” And then, attending 
only to the case of a system of free particles, he gives a definition, which, in 
the case of a single particle, is as follows :— 
Jacobi’s principal function S.—The equations of motion being as before 
Oe SE ee ee 
dé "dx? de” dy’ dé dz 
(where U is in general a function of x, y, z and ¢), then S is defined to be a 
complete solution of the partial differential equation 
dS, 1 dS\:  (dS8\’ , (dS\?\ 
tot a) +(3) + a) J gana 
A complete solution, it will be recollected, means a solution containing as 
many arbitrary constants as there are independent variables in the partial 
differential equations; in the present case, therefore, four arbitrary con- 
stants. But one of these constants may be taken to be a constant attached 
to the function S by mere addition, and which disappears from the dif- 
ferential coefficients, and it is only necessary to attend to the other three 
arbitrary constants. S is consequently a function of t,x,y,2, and of the 
arbitrary constants a, 3, y, satisfying the partial differential equation. And 
this being so, it is shown that the integrals of the problem are 
OB pp dS apy FB snip 
ae cenaliidins 46 cahow ¥; 
dS_, dS_d8_ 
du. ap ft? ay. 
where ),p,” are any other arbitrary constants, viz., the first three equa- 
tions give the intermediate integrals, and the last three equations give the 
final integrals of the problem. 
Jacobi proceeds to give an analogous definition of the principal function 
V as follows :— 
35. Jacobi’s principal function V.—First, when the condition of vis viva 
is satisfied. Here V is a complete solution of the partial differential 
equation 
1 GN a VA2s a Vint 
aa (Se) e i) +(%) }=u+a, 
where A is the constant of vis viva. The partial differential equation con- 
tains only three independent variables; and since as before one of the 
constants of the complete solution may be taken to be a constant attached 
to V by mere addition, and which disappears from the differential co- 
efficients, we may consider V as a function of ¢, x,y,z, and of the two con- 
stants of integration a and £. But V will of course also contain the 
constant h, which enters into the partial differential equation. The integrals 
of the problem are then shown to be 
