PARTIAL DIFFERENTIAL EQUATIONS. 
153 
when the single integral is taken round the boundary of the area over which the 
double integral is to extend. 
Further, X, Y are in this case the partial derivatives of a single function, 
§ 3. Let us consider the double integral 
| j(X dy dz + Y dz dx -f- Z dx dy), 
where X, Y, Z are functions of the independent variables x, y, z. It is known 
that this, taken over a closed surface under certain restrictions, is .equal to the triple 
integral 
J J J(0X/0x + 0Y /By + dZ/dz) dx dy dz 
taken over the space enclosed by that surface. 
Hence, if dX/dx + 0Y /dy -j- dZjdz = 0 identically the double integral taken over 
a closed surface vanishes, and taken over two open surfaces with the same boundary 
has the same value; that is to say, the value of the double integral depends on the 
values of x, y, z at the boundary only, and not, under certain restrictions, on the 
form of the surface enclosed by the boundary. 
By analogy we may call the element of the double integral a “ complete double 
differential,” or a “ complete bidifferential ” under these circumstances ; the condition 
that X dy dz -j- Y dzdx + Z dx dy may be a complete bidifferential is thus 
dX/dx -f- 0Y /dy -f- dZ/dz = 0. 
§ 4. A complete bidifferential may be expressed as a single term, such as du dv. 
For let u, v be two independent solutions of the equation 
x| + Y^ + Z^ = 0, 
dx- dy dz 
so that u = a, v = b are integrals of the system 
dxJX = dy/Y = dzjZ ; 
then 
X = 0 
6 being some multiplier, 
and 
5Q, v) y _ a d(tt. v ) 7 — 0 v ) 
d(y,z)' ~ d(z, x) ’ — d{x,y ’ 
0X 0Y 0Z 0(£l, u, v) 
dx dy dz d(x, y, z)‘ 
Since the last vanishes identically 6 is a function of u, v only ; a function w of u, v 
may be found, such that dw/du = 6, and thus 
_ 0Q, v ) _ d(w, v) 
— d(y, z) 5 — 9 ( 2 , x) ’ 
VOL. <JX(JV.— A. 
X 
„ _ d(w, v) 
~ 3(«. y) * 
