176 
MESSRS. A. E. H. LOVE AND F. B. PIDDUCK 
8. Integration of the Equation in Special Cases .—When n is a positive integer, the 
equation can be integrated. We write for a moment D for cj?u, and observe that, if 
D were a constant, the equation 
0 2 Z 2 n 0Z 
0(T 2 
+ 
- D 2 Z = 0 
would be a form of Riccati’s equation, and could be integrated in the form 
z = (I i-V -1 ( g<TDA + e ~ aDB ) , 
\cr Her: u 
where A and B are independent of a. Treating them as functions of u, we obtain the 
general primitive of the equation for Z in the form 
+ u) +f — 
<T j 
9. More General Integration. —Interpreting the variables r and s as the co-ordinates 
of a point in a plane, Riemann showed how to integrate the equation for Z when the 
values of this function and its first differential coefficients are given along an arc of a 
curve in the plane. If A" satisfies the “ adjoint ” equation 
c a Y 
dr ds 
f 
—11 
Y 
the integral 
J jv (A 
or L 
. + ^ 
cs r + s 
-h — 
cr 0s/ \r + s 
_ 1 [z (W 
ds r \ dr 
= 0 ,- 
nV 
r + sJ J _ 
dr ds, 
taken over any area in the plane, is equal to 
JI 
V 
CS ^ 
or c.s 
+ 
n 
r + s 
)Z 
■> 
CS / 
c V 
3r 0s 
—n 
\ 
or 
r + s 
dr ds. 
and therefore vanishes. It follows that the line-integral 
3Z nZ 
ds r + S/ 
* + z ( A 
nX 
7- + S 
taken round the boundary of the area vanishes. 
Let the values of Z and its first differential coefficients be given along an arc AC of a 
curve, and let P be a point which is not on the arc. Through P let lines PA and PC 
be drawn parallel to the axes of r and s, and let the area of integration be that bounded 
