LINE INTEGRALS AND GREEN’S THEOREM 225 
6. Prove that 
7. If u is a solution of Laplace’s equation, 
Chi chi 
cx 2 cy 2 
show that 
8. If u is a solution of Laplace’s equation, show that 
9. If u is a solution of Laplace’s equation, which is not a con 
stant, show that 
5. The Integral j Pdx + Qdy. 
C 
Theorem 1. Let P and Q be two functions which, together with the 
derivatives dP/cy and cQ/dx, are continuous within and on the bound 
ary of S. Let 
dP = dQ 
cy cx 
(1) 
at evei'y point of S. Let 2 be any region lying in S; the boundary C 
of 2 may coincide in part (or wholly) with that of S. Then 
Pdx -i- Qdy = 0, 
the integral being extended over the complete boundary of $ in the posi 
tive sense. 
The proof is given immediately by means of the relation (C) of 
§ 4, since the double integral has the value zero. 
Theorem 2. Conversely, if P and Q are continuous, together with 
dP/dy and cQ/dx, within and on the boundary of S, and if 
c