PARTIAL DIFFERENTIAL EQUATIONS. 
157 
dp dq. 
T P Y 9 (?^, *?) 
J-I -A-rs — v 
c(x ry x^) 
integral is d(u, v)/d(p, q), so that the integral becomes jj du dv. Its value will 
therefore only depend on the values of u, v, that is of x l} x. 2 . . . x n , at the boundary 
of the range of integration, and not on the form of the relations giving x x , x 2 . . . 
in terms of p, q, which define the particular surface over which the integral is taken. 
In this case we may write 
for all pairs of suffixes, the subject of integration in the last 
2 X„ d(x r , x s ) — d(u, v) 
r, s 
and call it a complete bidifferential. It is easily seen that the coefficients X satisfy 
the relations 
X,, Xy + X,, ~Xj S + X* X,; =0j ....... (1) 
ax,, ax*, 
dxi dx s 
+ = ° 
( 2 ), 
for all combinations of suffixes, where it is understood that the term X,//(.r„ x s ) may 
be also written X. sr d(x s , x r ), so that 
The conditions (2) are those which must be satisfied in order that the value of the 
double integral may depend only on the boundary. The difference of two values 
of the double integral, for which the same boundary is assumed, will be its value over 
a closed surface passing through the boundary curve, and this may be transformed 
into the triple integral 
, 0X ; , SXX , 
+ 4 - )dXi dx r dx s 
ox o dr- 
taken through the volume of any solid bounded by this closed surface. Hence this 
integral must vanish for any solid. By taking an infinitesimal solid, for every point 
of which all but X{, x r , x s are constant, we find the condition (2). 
The conditions (2) would be satisfied by an expression which was the sum of two 
or more complete bidifferentials, but (1) in general would not. 
§ 9. We next try to find whether these conditions are sufficient as well as 
necessary. Now all the coefficients X cannot vanish. Suppose that X 12 does not, 
then we have from (1) 
X„ = 
Xj, X, 8 - X u Xo,. 
• x 13 
(r, s — 3, 4 
n) 
and in virtue of these all the conditions (I) are satisfied. 
