( 732 ) 
lie gt heme 
A are ha, te ee ae 
The right-hand member of this formula remaining the same if 
we move the system along the path P, the proposition is proved. 
§ 27. A remarkable and well known theorem of JAcoBr is a direct 
consequence of our last proposition. If we have found a function 
R of the coordinates, satisfying the differential equation (33) and 
containing, besides an additive. constant, 8”—i—1 other arbitrary 
ok OR 
constants cj c, ete, the values of —, ae ete. will not change, 
EL, 0h 
while the system describes a path, perpendicular to the surface 
Beis ves, Sats NEO oe he ae 
The 8n—7z—1 equations of such a path will therefore be of the form 
R LR 
in Yi ee Mor LE ret pere ne NGE 
where vj, %e, etc. are constants. 
The total number of the constants c and 7 is 2 (3n—i—1);. this 
is just sufficient for the representation of every path that may be 
described with the chosen value of the energy. If we confine ourselves 
to fixed values of the constants c, and change those of the constants 7, 
we shall obtain all paths that are perpendicular to one and the 
same surface (41). One of these paths will be distinguished from 
the other by the value of the action along the parts of the path, 
lying between the surface (41) on one, and each of the surfaces 
KR (ey + dey, Co, Coy 2 2 Ni) 
A GER en el AT oe) = 
ve (cj, Ca, C3 4- d Coy eee oy == 0, 
on the other side. Indeed, by (42) the action along these parts is 
yadeg etc. 
By giving other values to the constants ¢, we shall change the 
surface (41) and we shall find the paths that are perpendicular to 
the new surface, 
— yder, 
